Principles of ElectrochemistrySecond Edition

Jin Koryta

Institute of Physiology,Czechoslovak Academy of Sciences, Prague

•Win Dvorak

Department of Physical Chemistry, Faculty of Science,Charles University, Prague

Ladislav Kavan

/. Heyrovsky Institute of Physical Chemistry and Electrochemistry,Czechoslovak Academy of Sciences, Prague

JOHN WILEY & SONSChichester • New York • Brisbane • Toronto • Singapore

Copyright © 1987, 1993 by John Wiley & Sons Ltd.Baffins Lane, Chichester,West Sussex PO19 1UD, England

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Library of Congress Cataloging-in-Publication Data

Koryta, Jifi.Principles of electrochemistry.—2nd ed. / Jin Koryta, Jin

Dvorak, Ladislav Kavan.p. cm.

Includes bibliographical references and index.ISBN 0 471 93713 4 : ISBN 0 471 93838 6 (pbk)1. Electrochemistry. I. Dvorak, Jin, II. Kavan, Ladislav.

III. Title.QD553.K69 1993541.37—dc20 92-24345

CIP

British Library Cataloguing in Publication Data

A catalogue record for this book is availablefrom the British Library

ISBN 0 471 93713 4 (cloth)ISBN 0 471 93838 6 (paper)

Typeset in Times 10/12 pt by The Universities Press (Belfast) Ltd.Printed and bound in Great Britain by Biddies Ltd, Guildford, Surrey

Contents

Preface to the First Edition xi

Preface to the Second Edition xv

Chapter 1 Equilibrium Properties of Electrolytes 11.1 Electrolytes: Elementary Concepts 11.1.1 Terminology 11.1.2 Electroneutrality and mean quantities 31.1.3 Non-ideal behaviour of electrolyte solutions 41.1.4 The Arrhenius theory of electrolytes 91.2 Structure of Solutions 131.2.1 Classification of solvents 131.2.2 Liquid structure 141.2.3 Ionsolvation 151.2.4 Ion association 231.3 Interionic Interactions 281.3.1 The Debye-Huckel limiting law 291.3.2 More rigorous Debye-Hiickel treatment of the activity

coefficient 341.3.3 The osmotic coefficient 381.3.4 Advanced theory of activity coefficients of electrolytes . . . . 381.3.5 Mixtures of strong electrolytes 411.3.6 Methods of measuring activity coefficients 441.4 Acids and Bases 451.4.1 Definitions 451.4.2 Solvents and self-ionization 471.4.3 Solutions of acids and bases 501.4.4 Generalization of the concept of acids and bases 591.4.5 Correlation of the properties of electrolytes in various solvents 611.4.6 The acidity scale 631.4.7 Acid-base indicators 651.5 Special Cases of Electrolytic Systems 691.5.1 Sparingly soluble electrolytes 69

v

VI

1.5.2 Ampholytes 701.5.3 Polyelectrolytes 73

Chapter 2 Transport Processes in Electrolyte Systems 792.1 Irreversible Processes 792.2 Common Properties of the Fluxes of Thermodynamic Quantities 812.3 Production of Entropy, the Driving Forces of Transport

Phenomena 842.4 Conduction of Electricity in Electrolytes 872.4.1 Classification of conductors 872.4.2 Conductivity of electrolytes 902.4.3 Interionic forces and conductivity 932.4.4 The Wien and Debye-Falkenhagen effects 982.4.5 Conductometry 1002.4.6 Transport numbers 1012.5 Diffusion and Migration in Electrolyte Solutions 1042.5.1 The time dependence of diffusion 1052.5.2 Simultaneous diffusion and migration 1102.5.3 The diffusion potential and the liquid junction potential . . . I l l2.5.4 The diffusion coefficient in electrolyte solutions 1152.5.5 Methods of measurement of diffusion coefficients 1182.6 The Mechanism of Ion Transport in Solutions, Solids, Melts, and

Polymers 1202.6.1 Transport in solution 1212.6.2 Transport in solids 1242.6.3 Transport in melts 1272.6.4 Ion transport in polymers 1282.7 Transport in a flowing liquid 1342.7.1 Basic concepts 1342.7.2 The theory of convective diffusion 1362.7.3 The mass transfer approach to convective diffusion 141

Chapter 3 Equilibria of Charge Transfer in HeterogeneousElectrochemical Systems 144

3.1 Structure and Electrical Properties of Interfacial Regions . . . . 1443.1.1 Classification of electrical potentials at interfaces 1453.1.2 The Galvani potential difference 1483.1.3 The Volta potential difference 1533.1.4 The EMF of galvanic cells 1573.1.5 The electrode potential 1633.2 Reversible Electrodes 1693.2.1 Electrodes of the first kind 1703.2.2 Electrodes of the second kind 175

Vll

3.2.3 Oxidation-reduction electrodes 1773.2.4 The additivity of electrode potentials, disproportionation . . . 1803.2.5 Organic redox electrodes 1823.2.6 Electrode potentials in non-aqueous media 1843.2.7 Potentials at the interface of two immiscible electrolyte

solutions 1883.3 Potentiometry 1913.3.1 The principle of measurement of the EMF 1913.3.2 Measurement of pH 1923.3.3 Measurement of activity coefficients 1953.3.4 Measurement of dissociation constants 195

Chapter 4 The Electrical Double Layer 1984.1 General Properties 1984.2 Electrocapillarity 2034.3 Structure of the Electrical Double Layer 2134.3.1 Diffuse electrical layer 2144.3.2 Compact electrical layer 2174.3.3 Adsorption of electroneutral molecules 2244.4 Methods of the Electrical Double-layer Study 2314.5 The Electrical Double Layer at the Electrolyte-Non-metallic

Phase Interface 2354.5.1 Semiconductor-electrolyte interfaces 2354.5.2 Interfaces between two electrolytes 2404.5.3 Electrokinetic phenomena 242

Chapter 5 Processes in Heterogeneous Electrochemical Systems . . 2455.1 Basic Concepts and Definitions 2455.2 Elementary outline for simple electrode reactions 2535.2.1 Formal approach 2535.2.2 The phenomenological theory of the electrode reaction . . . . 2545.3 The Theory of Electron Transfer 2665.3.1 The elementary step in electron transfer 2665.3.2 The effect of the electrical double-layer structure on the rate of

the electrode reaction 2745.4 Transport in Electrode Processes 2795.4.1 Material flux and the rate of electrode processes 2795.4.2 Analysis of polarization curves (voltammograms) 2845.4.3 Potential-sweep voltammetry 2885.4.4 The concentration overpotential 2895.5 Methods and Materials 2905.5.1 The ohmic electrical potential difference 2915.5.2 Transition and steady-state methods 293

V l l l

5.5.3 Periodic methods 3015.5.4 Coulometry 3035.5.5 Electrode materials and surface treatment 3055.5.6 Non-electrochemical methods 3285.6 Chemical Reactions in Electrode Processes 3445.6.1 Classification 3455.6.2 Equilibrium of chemical reactions 3465.6.3 Chemical volume reactions 3475.6.4 Surface reactions 3505.7 Adsorption and Electrode Processes 3525.7.1 Electrocatalysis 3525.7.2 Inhibition of electrode processes 3615.8 Deposition and Oxidation of Metals 3685.8.1 Deposition of a metal on a foreign substrate 3695.8.2 Electrocrystallization on an identical metal substrate 3725.8.3 Anodic oxidation of metals 3775.8.4 Mixed potentials and corrosion phenomena 3815.9 Organic Electrochemistry 3845.10 Photoelectrochemistry 3905.10.1 Classification of photoelectrochemical phenomena 3905.10.2 Electrochemical photoemission 3925.10.3 Homogeneous photoredox reactions and photogalvanic effects 3935.10.4 Semiconductor photoelectrochemistry and photovoltaic effects 3975.10.5 Sensitization of semiconductor electrodes 4035.10.6 Photoelectrochemical solar energy conversion 406

Chapter 6 Membrane Electrochemistry and Bioelectrochemistry . . 4106.1 Basic Concepts and Definitions 4106.1.1 Classification of membranes 4116.1.2 Membrane potentials 4116.2 Ion-exchanger Membranes 4156.2.1 Classification of porous membranes 4156.2.2 The potential of ion-exchanger membranes 4176.2.3 Transport through a fine-pore membrane 4196.3 Ion-selective Electrodes 4256.3.1 Liquid-membrane ion-selective electrodes 4256.3.2 Ion-selective electrodes with fixed ion-exchanger sites . . . . 4286.3.3 Calibration of ion-selective electrodes 4316.3.4 Biosensors and other composite systems 4316.4 Biological Membranes 4336.4.1 Composition of biological membranes 4346.4.2 The structure of biological membranes 4386.4.3 Experimental models of biological membranes 4396.4.4 Membrane transport 4426.5 Examples of Biological Membrane Processes 454

IX

6.5.1 Processes in the cells of excitable tissues 4546.5.2 Membrane principles of bioenergetics 464

Appendix A Recalculation Formulae for Concentrations andActivity Coefficients 473

Appendix В List of Symbols 474

Index 477

Preface to the First Edition

Although electrochemistry has become increasingly important in societyand in science the proportion of physical chemistry textbooks devoted toelectrochemistry has declined both in extent and in quality (with notableexceptions, e.g. W. J. Moore's Physical Chemistry).

As recent books dealing with electrochemistry have mainly been ad-dressed to the specialist it has seemed appropriate to prepare a textbook ofelectrochemistry which assumes a knowledge of basic physical chemistry atthe undergraduate level. Thus, the present text will benefit the moreadvanced undergraduate and postgraduate students and research workersspecializing in physical chemistry, biology, materials science and theirapplications. An attempt has been made to include as much material aspossible so that the book becomes a starting point for the study ofmonographs and original papers.

Monographs and reviews (mainly published after 1970) pertaining toindividual sections of the book are quoted at the end of each section. Manyreviews have appeared in monographic series, namely:

Advances in Electrochemistry and Electrochemical Engineering (Eds P.Delahay, H. Gerischer and C. W. Tobias), Wiley-Interscience, NewYork, published since 1961, abbreviation in References AE.

Electroanalytical Chemistry (Ed. A. J. Bard), M. Dekker, New York,published since 1966.

Modern Aspects of Electrochemistry (Eds J. O'M. Bockris, В. Е. Conwayand coworkers), Butterworths, London, later Plenum Press, New York,published since 1954, abbreviation MAE.

Electrochemical compendia include:

The Encyclopedia of Electrochemistry (Ed. C. A. Hempel), Reinhold, NewYork, 1961.

Comprehensive Treatise of Electrochemistry (Eds J. O'M. Bockris, В. Е.Conway, E. Yeager and coworkers), 10 volumes, Plenum Press, 1980-1985, abbreviation CTE.

Electrochemistry of Elements (Ed. A. J. Bard), M. Dekker, New York, amultivolume series published since 1973.

xi

Xll

Physical Chemistry. An Advanced Treatise (Eds H. Eyring, D. Hendersonand W. Jost), Vol. IXA,B, Electrochemistry, Academic Press, NewYork, 1970, abbreviation PChAT.

Hibbert, D. B. and A. M. James, Dictionary of Electrochemistry,Macmillan, London, 1984.

There are several more recent textbooks, namely:

Bockris, J. O'M. and A. K. N. Reddy, Modern Electrochemistry, PlenumPress, New York, 1970.

Hertz, H. G., Electrochemistry—A Reformulation of Basic Principles,Springer-Verlag, Berlin, 1980.

Besson, J., Precis de Thermodynamique et Cinetique Electrochimique,Ellipses, Paris, 1984, and an introductory text.

Koryta, J., Ionsy Electrodesy and Membranes, 2nd Ed., John Wiley & Sons,Chichester, 1991.

Rieger, P. H., Electrochemistry, Prentice-Hall, Englewood Cliffs, N.J.,1987.

The more important data compilations are:

Conway, В. Е., Electrochemical Data, Elsevier, Amsterdam, 1952.CRC Handbook of Chemistry and Physics (Ed. R. C. Weast), CRC Press,

Boca Raton, 1985.CRC Handbook Series in Inorganic Electrochemistry (Eds L. Meites, P.

Zuman, E. B. Rupp and A. Narayanan), CRC Press, Boca Raton, amultivolume series published since 1980.

CRC Handbook Series in Organic Electrochemistry (Eds L. Meites and P.Zuman), CRC Press, Boca Raton, a multivolume series published since1977.

Horvath, A. L., Handbook of Aqueous Electrolyte Solutions, PhysicalProperties, Estimation and Correlation Methods, Ellis Horwood, Chiches-ter, 1985.

Oxidation-Reduction Potentials in Aqueous Solutions (Eds A. J. Bard, J.Jordan and R. Parsons), Blackwell, Oxford, 1986.

Parsons, R., Handbook of Electrochemical Data, Butterworths, London,1959.

Perrin, D. D., Dissociation Constants of Inorganic Acids and Bases inAqueous Solutions, Butterworths, London, 1969.

Standard Potentials in Aqueous Solutions (Eds A. J. Bard, R. Parsons andJ. Jordan), M. Dekker, New York, 1985.The present authors, together with the late (Miss) Dr V. Bohackova,

published their Electrochemistry, Methuen, London, in 1970. In spite of thefavourable attitude of the readers, reviewers and publishers to that book(German, Russian, Polish, and Czech editions have appeared since then) wenow consider it out of date and therefore present a text which has beenlargely rewritten. In particular we have stressed modern electrochemical

хш

materials (electrolytes, electrodes, non-aqueous electrochemistry in gene-ral), up-to-date charge transfer theory and biological aspects of electro-chemistry. On the other hand, the presentation of electrochemical methodsis quite short as the reader has access to excellent monographs on thesubject (see page 301).

The Czech manuscript has been kindly translated by Dr M. Hyman-Stulikova. We are much indebted to the late Dr A Ryvolova, Mrs M.Kozlova and Mrs D. Tumova for their expert help in preparing themanuscript. Professor E. Budevski, Dr J. Ludvik, Dr L. Novotny and Dr J.Weber have supplied excellent photographs and drawings.

Dr K. Janacek, Dr L. Kavan, Dr K. Micka, Dr P. Novak, Dr Z. Samecand Dr J. Weber read individual chapters of the manuscript and madevaluable comments and suggestions for improving the book. Dr L. Kavan isthe author of the section on non-electrochemical methods (pages 319 to329).

We are also grateful to Professor V. Pokorny, Vice-president of theCzechoslovak Academy of Sciences and chairman of the Editorial Councilof the Academy, for his support.

Lastly we would like to mention with devotion our teachers, the lateProfessor J. Heyrovsky and the late Professor R. Brdicka, for theinspiration we received from them for our research and teaching ofelectrochemistry, and our colleague and friend, the late Dr V. Bohackova,for all her assistance in the past.

Prague, March 1986 Jifi KorytaJin Dvorak

Preface to the Second Edition

The new edition of Principles of Electrochemistry has been considerablyextended by a number of new sections, particularly dealing with 'electro-chemical material science' (ion and electron conducting polymers, chemicallymodified electrodes), photoelectrochemistry, stochastic processes, new asp-ects of ion transfer across biological membranes, biosensors, etc. In view ofthis extension of the book we asked Dr Ladislav Kavan (the author of thesection on non-electrochemical methods in the first edition) to contribute asa co-author discussing many of these topics. On the other hand it has beennecessary to become less concerned with some of the 'classical' topics thedetails of which are of limited importance for the reader.

Dr Karel Micka of the J. Heyrovsky Institute of Physical Chemistry andElectrochemistry has revised very thoroughly the language of the originaltext as well as of the new manuscript. He has also made many extremelyuseful suggestions for amending factual errors and improving the accuracyof many statements throughout the whole text. We are further muchindebted to Prof. Michael Gratzel and Dr Nicolas Vlachopoulos, FederalPolytechnics, Lausanne, for valuable suggestions to the manuscript.

During the preparation of the second edition Professor Jiff Dvorak diedafter a serious illness on 27 February 1992. We shall always remember hisscientific effort and his human qualities.

Prague, May 1992 Jifi Koryta

xv

Chapter 1

Equilibrium Properties ofElectrolytes

Substances are frequently spoken of as being electro-negative, or electro-positive,according as they go under the supposed influence of direct attraction to the positiveor negative pole. But these terms are much too significant for the use for which Ishould have to put them; for though the meanings are perhaps right, they are onlyhypothetical, and may be wrong; and then, through a very imperceptible, but stillvery dangerous, because continual, influence, they do great injury to science, bycontracting and limiting the habitual views of those engaged in pursuing it. I proposeto distinguish such bodies by calling those anions which go to the anode of thedecomposing body; and those passing to the cathode, cations•; and when I have theoccasion to speak of these together, I shall call them ions. Thus, the chloride of leadis an electrolyte, and when electrolysed evolves two ions, chlorine and lead, theformer being an anion, and the latter a cation.

M. Faraday, 1834

1.1 Electrolytes: Elementary Concepts

1.1.1 Terminology

A substance present in solution or in a melt which is at least partly in theform of charged species—ions—is called an electrolyte. The decompositionof electroneutral molecules to form electrically charged ions is termedelectrolytic dissociation. Ions with a positive charge are called cations; thosewith a negative charge are termed anions. Ions move in an electric field as aresult of their charge—cations towards the cathode, anions to the anode.The cathode is considered to be that electrode through which negativecharge, i.e. electrons, enters a heterogeneous electrochemical system(electrolytic cell, galvanic cell). Electrons leave the system through theanode. Thus, in the presence of current flow, reduction always occurs at thecathode and oxidation at the anode. In the strictest sense, in the absence ofcurrent passage the concepts of anode and cathode lose their meaning. Allthese terms were introduced in the thirties of the last century by M.Faraday.

R. Clausius (1857) demonstrated the presence of ions in solutions andverified the validity of Ohm's law down to very low voltages (by electrolysis

1

of a solution with direct current and unpolarizable electrodes). Up until thattime, it was generally accepted that ions are formed only under theinfluence of an electric field leading to current flow through the solution.

The electrical conductivity of electrolyte solutions was measured at thevery beginning of electrochemistry. The resistance of a conductor R is theproportionality constant between the applied voltage U and the current /passing through the conductor. It is thus the constant in the equationU = RI, known as Ohm's law. The reciprocal of the resistance is termed theconductance. The resistance and conductance depend on the material fromwhich the conductor is made and also on the length L and cross-section S ofthe conductor. If the resistance is recalculated to unit length and unitcross-section of the conductor, the quantity p = RS/L is obtained, termedthe resistivity. For conductors consisting of a solid substance (metals, solidelectrolytes) or single component liquids, this quantity is a characteristic ofthe particular substance. In solutions, however, the resistivity and theconductivity K = l/p are also dependent on the electrolyte concentration c.In fact, even the quantity obtained by recalculation of the conductivity tounit concentration, A = K/C, termed the molar conductivity, is not inde-pendent of the electrolyte concentration and is thus not a material constant,characterizing the given electrolyte. Only the limiting value at very lowconcentrations, called the limiting molar conductivity A0, is such a quantity.

A study of the concentration dependence of the molar conductivity,carried out by a number of authors, primarily F. W. G. Kohlrausch and W.Ostwald, revealed that these dependences are of two types (see Fig. 2.5)and thus, apparently, there are two types of electrolytes. Those that arefully dissociated so that their molecules are not present in the solution arecalled strong electrolytes, while those that dissociate incompletely are weakelectrolytes. Ions as well as molecules are present in solution of a weakelectrolyte at finite dilution. However, this distinction is not very accurateas, at higher concentration, the strong electrolytes associate formingion pairs (see Section 1.2.4).

Thus, in weak electrolytes, molecules can exist in a similar way as innon-electrolytes—a molecule is considered to be an electrically neutralspecies consisting of atoms bonded together so strongly that this species canbe studied as an independent entity. In contrast to the molecules ofnon-electrolytes, the molecules of weak electrolytes contain at least onebond with a partly ionic character. Strong electrolytes do not formmolecules in this sense. Here the bond between the cation and the anion isprimarily ionic in character and the corresponding chemical formularepresents only a formal molecule; nonetheless, this formula correctlydescribes the composition of the ionic crystal of the given strong electrolyte.

The first theory of solutions of weak electrolytes was formulated in 1887by S. Arrhenius (see Section 1.1.4). If the molar conductivity is introducedinto the equations following from Arrhenius' concepts of weak electrolytes,Eq. (2.4.17) is obtained, known as the Ostwald dilution law; this equation

provides a good description of one of these types of concentrationdependence of the molar conductivity. The second type was described byKohlrausch using the empirical equation (2.4.15), which was later theoreti-cally interpreted by P. Debye and E. Hiickel on the basis of concepts of theactivity coefficients of ions in solutions of completely dissociated el-ectrolytes, and considerably improved by L. Onsager. An electrolyte can beclassified as strong or weak according to whether its behaviour can bedescribed by the Ostwald or Kohlrausch equation. Similarly, the 'strength'of an electrolyte can be estimated on the basis of the van't Hoff coefficient(see Section 1.1.4).

1.1.2 Electroneutrality and mean quantities

Prior to dissolution, the ion-forming molecules have an overall electriccharge of zero. Thus, a homogeneous liquid system also has zero chargeeven though it contains charged species. In solution, the number of positiveelementary charges on the cations equals the number of negative charges ofthe anions. If a system contains s different ions with molality m,(concentrations c, or mole fractions xt can also be employed), each bearing2/ elementary charges, then the equation

0 (1.1.1)1 = 1

called the electroneutrality condition, is valid on a macroscopic scale forevery homogeneous part of the system but not for the boundary betweentwo phases (see Chapter 4).

From the physical point of view there cannot exist, under equilibriumconditions, a measurable excess of charge in the bulk of an electrolytesolution. By electrostatic repulsion this charge would be dragged to thephase boundary where it would be the source of a strong electric field in thevicinity of the phase. This point will be discussed in Section 3.1.3.

In Eq. (1.1.1), as elsewhere below, z, is a dimensionless number (thecharge of species i related to the charge of a proton, i.e. the charge numberof the ion) with sign zt > 0 for cations and z, < 0 for anions.

The electroneutrality condition decreases the number of independentvariables in the system by one; these variables correspond to componentswhose concentration can be varied independently. In general, however,a number of further conditions must be maintained (e.g. stoichiometryand the dissociation equilibrium condition). In addition, because of theelectroneutrality condition, the contributions of the anion and cation to anumber of solution properties of the electrolyte cannot be separated (e.g.electrical conductivity, diffusion coefficient and decrease in vapour pressure)without assumptions about individual particles. Consequently, mean valueshave been defined for a number of cases.

For example, the molality can be expressed for an electrolyte as a whole,mx\ the amount of substance ('number of moles') is expressed in moles of

formula units as if the electrolyte were not dissociated. For a strongelectrolyte whose formula unit contains v+ cations and v_ anions, i.e. atotal of v = v+ + v_ ions, the molalities of the ions are related to the totalmolality by a simple relationship, ra+ = v+ml and m_ = v_mY. The meanmolality is then

m± = (ml+mv-)l/v = m,« + vr - ) 1 / v (1.1.2)

The mean molality values m± (moles per kilogram), mole fractions x±

(dimensionless number) and concentrations c± (moles per cubic decimetre)are related by equations similar to those for non-electrolytes (see AppendixA).

1.1.3 Non-ideal behaviour of electrolyte solutions

The chemical potential is encountered in electrochemistry in connectionwith the components of both solutions and gases. The chemical potential \i{

of component / is defined as the partial molar Gibbs energy of the system,i.e. the partial derivative of the Gibbs energy G with respect to the amountof substance nt of component i at constant pressure, temperature andamounts of all the other components except the ith. Consider that thesystem does not exchange matter with its environment but only energy inthe form of heat and volume work. From this definition it follows for areversible isothermal change of the pressure of one mole of an ideal gasfrom the reference value prcf to the actual value /?act that

A*act-^ef=/^ln— (1.1.3)Prcf

which is usually written in the form

\i = /i° + RT Inp (1.1.4)

where p is the dimensionless pressure ratio pact/pref' The reference state istaken as the state at the given temperature and at a pressure of 105 Pa. Thedimensionless pressure p is therefore expressed as multiples of thisreference pressure. Term jU° has the significance of the chemical potential ofthe gas at a pressure equal to the standard pressure, p = 1, and is termedthe standard chemical potential. This significance of quantities fj,° and pshould be recalled, e.g. when substituting pressure values into the Nernstequation for gas electrodes (see Section 3.2); if the value of the actualpressure in some arbitrary units were substituted (e.g. in pounds per squareinch), this would affect the value of the standard electrode potential.

The chemical potential //, of the components of an ideal mixture of liquids(the components of an ideal mixture of liquids obey the Raoult law over thewhole range of mole fractions and are completely miscible) is

^1 = pT + RT In Xi (1.1.5)

The standard term y,* is the chemical potential of the pure component i (i.e.when xt = 1) at the temperature of the system and the correspondingsaturated vapour pressure. According to the Raoult law, in an ideal mixturethe partial pressure of each component above the liquid is proportional toits mole fraction in the liquid,

Pi=P°Xi (1.1.6)

where the proportionality constant p° is the vapour pressure above the puresubstance.

In a general case of a mixture, no component takes preference and thestandard state is that of the pure component. In solutions, however, onecomponent, termed the solvent, is treated differently from the others, calledsolutes. Dilute solutions occupy a special position, as the solvent is presentin a large excess. The quantities pertaining to the solvent are denoted by thesubscript 0 and those of the solute by the subscript 1. For xl->0 and xo-*l,Po = Po a nd Pi — kiXi. Equation (1.1.5) is again valid for the chemicalpotentials of both components. The standard chemical potential of thesolvent is defined in the same way as the standard chemical potential of thecomponent of an ideal mixture, the standard state being that of the puresolvent. The standard chemical potential of the dissolved component juf isthe chemical potential of that pure component in the physically unattainablestate corresponding to linear extrapolation of the behaviour of thiscomponent according to Henry's law up to point xx = 1 at the temperatureof the mixture T and at pressure p = klt which is the proportionalityconstant of Henry's law.

For a solution of a non-volatile substance (e.g. a solid) in a liquid thevapour pressure of the solute can be neglected. The reference state for sucha substance is usually its very dilute solution—in the limiting case aninfinitely dilute solution—which has identical properties with an idealsolution and is thus useful, especially for introducing activity coefficients(see Sections 1.1.4 and 1.3). The standard chemical potential of such asolute is defined as

A*i = Km (pi-RT Inxt) (1.1.7)JC(j-»l

where yl is the chemical potential of the solute, xx its mole fraction and x{)

the mole fraction of the solvent.In the subsequent text, wherever possible, the quantities jU° and pf will

not be distinguished by separate symbols: only the symbol $ will beemployed.

In real mixtures and solutions, the chemical potential ceases to be a linearfunction of the logarithm of the partial pressure or mole fraction.Consequently, a different approach is usually adopted. The simple form ofthe equations derived for ideal systems is retained for real systems, but adifferent quantity a, called the activity (or fugacity for real gases), is

introduced. Imagine that the dissolved species are less 'active' than wouldcorrespond to their concentration, as if some sort of loss' of the giveninteraction were involved. The activity is related to the chemical potentialby the relationship

[i^tf + RTlna, (1.1.8)

As in electrochemical investigations low pressures are usually employed, theanalogy of activity for the gaseous state, the fugacity, will not be introducedin the present book.

Electrolyte solutions differ from solutions of uncharged species in theirgreater tendency to behave non-ideally. This is a result of differences in theforces producing the deviation from ideality, i.e. the forces of interactionbetween particles of the dissolved substances. In non-electrolytes, these areshort-range forces (non-bonding interaction forces); in electrolytes, theseare electrostatic forces whose relatively greater range is given by Coulomb'slaw. Consider the process of concentrating both electrolyte and non-electrolyte solutions. If the process starts with infinitely dilute solutions,then their initial behaviour will be ideal; with increasing concentrationcoulombic interactions and at still higher concentrations, van der Waalsnon-bonding interactions and dipole-dipole interactions will become impor-tant. Thus, non-ideal behaviour must be considered for electrolyte solutionsat much lower concentrations than for non-electrolyte solutions. 'Respectingnon-ideal behaviour' means replacing the mole fractions, molalities andmolar concentrations by the corresponding activities in all the thermo-dynamic relationships. For example, in an aqueous solution with a molarconcentration of 10~3 mol • dm~3, sodium chloride has an activity of0.967 x 10~3. Non-electrolyte solutions retain their ideal properties up toconcentrations that may be as much as two orders of magnitude higher, asillustrated in Fig. 1.1.

Thus, the deviation in the behaviour of electrolyte solutions from theideal depends on the composition of the solution, and the activity of thecomponents is a function of their mole fractions. For practical reasons, theform of this function has been defined in the simplest way possible:

ax = yxx (1.1.9)

where the quantity yx is termed the activity coefficient (the significance ofthe subscript x will be considered later). However, the complicationsconnected with solution non-ideality have not been removed but onlytransferred to the activity coefficient, which is also a function of theconcentration. The form of this function can be found either theoretically(the theory has been quite successful for electrolyte solutions, see Section1.3) or empirically. Practical calculations can be based on one of thetheoretical or semiempirical equations for the activity coefficient (for thesimple ions, the activity coefficient values are tabulated); the activitycoefficient is then multiplied by the concentration and the activity thus

-2 -1 0Log of molality

Fig. 1.1 The activity coefficient y of a non-electrolyte and mean activity coefficients y± of

electrolytes as functions of molality

obtained is substituted into a simple 'ideal' equation (e.g. the law of massaction for chemical equilibrium).

Activity ax is termed the rational activity and coefficient yx is the rationalactivity coefficient. This activity is not directly given by the ratio of thefugacities, as it is for gases, but appears nonetheless to be the best meansfrom a thermodynamic point of view for description of the behaviour of realsolutions. The rational activity corresponds to the mole fraction for idealsolutions (hence the subscript x). Both ax and yx are dimensionlessnumbers.

In practical electrochemistry, however, the molality m or molar con-centration c is used more often than the mole fraction. Thus, the molalactivity amy molal activity coefficient ym) molar activity ac and molar activitycoefficient yc are introduced. The adjective 'molal' is sometimes replaced by'practical'.

The following equations provide definitions for these quantities:

Yi,m o> lim Yi.m =0

lim Yi,c =0

(1.1.10)

The standard states are selected asm? = l mol • kg"1 and c? = 1 mol • dm 3.In this convention, the ratio ra//m° is numerically identical with the actualmolality (expressed in units of moles per kilogram). This is, however, the

dimensionless relative molality, in the same way that pressure p in Eq.(1.1.4) is the dimensionless relative pressure. The ratio cjc® is analogous.The symbols ra—»0 and c—>0 in the last two equations indicate that themolalities or concentrations of all the components except the solvent aresmall.

Because of the electroneutrality condition, the individual ion activitiesand activity coefficients cannot be measured without additional extrather-modynamic assumptions (Section 1.3). Thus, mean quantities are defined fordissolved electrolytes, for all concentration scales. E.g., for a solution of asingle strong binary electrolyte as

a± = (£ix+fl--)1/v, Y± = ( r : + r - ) 1 / v (1.1.11)

The numerical values of the activity coefficients yx, ym and yc (and also ofthe activities ax, am and ac) are different (for the recalculation formulae seeAppendix A). Obviously, for the limiting case (for a very dilute solution)

y±,* = y±,m = y±.c«i (1.1.12)The activity coefficient of the solvent remains close to unity up to quite

high electrolyte concentrations; e.g. the activity coefficient for water in anaqueous solution of 2 M KC1 at 25°C equals y0>x = 1.004, while the value forpotassium chloride in this solution is y±>x = 0.614, indicating a quite largedeviation from the ideal behaviour. Thus, the activity coefficient of thesolvent is not a suitable characteristic of the real behaviour of solutions ofelectrolytes. If the deviation from ideal behaviour is to be expressed interms of quantities connected with the solvent, then the osmotic coefficient isemployed. The osmotic pressure of the system is denoted as JZ and thehypothetical osmotic pressure of a solution with the same composition thatwould behave ideally as JT*. The equations for the osmotic pressures JZ andJZ* are obtained from the equilibrium condition of the pure solvent and ofthe solution. Under equilibrium conditions the chemical potential of thepure solvent, which is equal to the standard chemical potential at thepressure /?, is equal to the chemical potential of the solvent in the solutionunder the osmotic pressure JZ,

li%(T, p) = iio(T, p + JZ) = iil(Ty p + jz) + RT\n a0 (1.1.13)

where a0 is the activity of the solvent (the activity of the pure solvent isunity). As approximately

p + Jt)- l*°o(T, p) = VOJZ (1.1.14)

we obtain for the osmotic pressure

JZ= \na0 (1.1.15)

where v0 is the molar volume of the solvent. For a dilute solutionIn a0 = In x0 = In (1 - E *,-) ~ - E xt•• — Mo E mh giving for the ideal osmotic

pressure (Mo is the relative molecular mass of the solvent)

**=^2> (1.1.16)

and in terms of molar concentration c of a single electrolyte dissociating intov ions

JT* = VRTC (1.1.17)

The ratio JI/JI* (which is experimentally measurable) is termed the molalosmotic coefficient

The rational osmotic coefficient is defined by the equation

In a0 =<t>x\nxQ (1.1.19)

For a solution of a single electrolyte, the relationship between the meanactivity coefficient and the osmotic coefficient is given by the equation

Inrm

= -(l-ct>m)-\ ( l -4>m , )dlnm' (1.1.20)

following from the definitions and from the Gibbs-Duhem equation.In view of the electrostatic nature of forces that primarily lead to

deviation of the behaviour of electrolyte solutions from the ideal, theactivity coefficient of electrolytes must depend on the electric charge of allthe ions present. G. N. Lewis, M. Randall and J. N. Br0nsted foundexperimentally that this dependence for dilute solutions is described quiteadequately by the relationship

log y±=Az+z_Vi (1.1.21)

in which the constant A for 25° and water has a value close to 0.5dm3/2 • mol~1/2. Quantity /, called the ionic strength, describes the electro-static effect of individual ionic species by the equation

/ = JXc,zf (1.1.22)i

(In fact, the symbol Ic should be used, as the molality ionic strength lm canbe defined analogously; in dilute aqueous solutions, however, values of cand m, and thus also Ic and /m, become identical.) Equation (1.1.21) waslater derived theoretically and is called the Debye-Hiickel limiting law. Itwill be discussed in greater detail in Section 1.3.1.

1.1.4 The A rrhenius theory of electrolytes

At the end of the last century S. Arrhenius formulated the firstquantitative theory describing the behaviour of weak electrolytes. The

10

existence of ions in solution had already been demonstrated at that time,but very little was known of the structure of solutions and the solvent wasregarded as an inert medium. Similarly, the concepts of the activity andactivity coefficient were not employed. Electrochemistry was limited toaqueous solutions. However, the basis of classical thermodynamics wasalready formulated (by J. W. Gibbs, W. Thomson and H. v. Helmholtz)and electrolyte solutions had also been investigated thermodynamicallyespecially by means of cryoscopic, osmometric and vapour pressuremeasurements.

Van't Hoff introduced the correction factor i for electrolyte solutions; themeasured quantity (e.g. the osmotic pressure, Jt) must be divided by thisfactor to obtain agreement with the theory of dilute solutions of non-electrolytes (jz/i = RTc). For the dilute solutions of some electrolytes (nowcalled strong), this factor approaches small integers. Thus, for a dilutesodium chloride solution with concentration c, an osmotic pressure of 2RTcwas always measured, which could readily be explained by the fact that thesolution, in fact, actually contains twice the number of species correspond-ing to concentration c calculated in the usual manner from the weighedamount of substance dissolved in the solution. Small deviations fromintegral numbers were attributed to experimental errors (they are nowattributed to the effect of the activity coefficient).

For other electrolytes, now termed weak, factor / has non-integral valuesdepending on the overall electrolyte concentration. This fact was explainedby Arrhenius in terms of a reversible dissociation reaction, whose equi-librium state is described by the law of mass action.

A weak electrolyte Bv+Av_ dissociates in solution to yield v ionsconsisting of v+ cations Bz+ and v_ anions Az~,

Bv+Av _<=* v+Bz+ + v_Az~ (1.1.23)

Thus the magnitude of the constant called the thermodynamic or realdissociation constant,

° ^ (1.1.24)

is a measure of the 'strength of the electrolyte'. The smaller its value, theweaker the electrolyte. The activity can be replaced by the concentrationsaccording to Eq. (1.1.10), yielding

rp>z+iv+r A Z - I V v+ v_ v+ v_

K —F^—A—T = K (1.1.25)Bv+Av_ yBZ yBA

where

K'=-[Bv Av_]

11

is called the apparent dissociation constant. Constant K depends on thetemperature; the dependence on the pressure is usually neglected asequilibria in the condensed phase are involved. Constant K' also dependson the ionic strength and increases with increasing ionic strength, as followsfrom substitution of the limiting relationship (1.1.21) into Eq. (1.1.25). Forsimplicity, consider monovalent ions, that is v+ = v_ = 1, so that_log yB =log yA = ~AV7and log yBA = 0. Obviously, then, yB = yA= 10°sV/, yBA = 1and substitution and rearrangement yield

/T = / aO v 7 (1.1.27)

It should be noted that the activity appearing in the dissociation constantK is the dimensionless relative activity, and constant K' contains thedimensionless relative concentration or molality terms. Constants K and K'are thus also dimensionless. However, their numerical values correspond tothe units selected for the standard state, i.e. moles per cubic decimetre ormoles per kilogram.

Because the dissociation constants for various electrolytes differ byseveral order of magnitude, the following definition

pK=-\ogK\ pK' =-log K' (1.1.28)

is introduced to characterize the electrolyte strength in terms of alogarithmic quantity. Operator/? appears frequently in electrochemistry andis equal to the log operator times —1 (i.e. px = —logjc).

The degree of dissociation a is the equilibrium degree of conversion, i.e.the fraction of the number of molecules originally present that dissociated atthe given concentration. The degree of dissociation depends directly on thegiven dissociation constant. Obviously a = [B2+]/v+c = [A2~]/v_c,[Bv+Av_] = c(l — a) and the dissociation constant is then given as

~ -1 — a (1.1.29)

The most common electrolytes are uni-univalent (v = 2, v+ = v_ = 1), forwhich

The relationship for a follows:

In moderately diluted solutions, i.e. for concentrations fulfilling thecondition, c» Kf,

a^(K'/c)l/2«l (1.1.32)

12

Id7 165

Concentration101

Fig. 1.2 Dependence of the dissociation degree a of a weekelectrolyte on molar concentration c for different values of the

apparent dissociation constant K' (indicated at each curve)

In the limiting case (readily obtained by differentiation of the numeratorand denominator with respect to c) it holds that ar-»l for c—»0, i.e. eachweak electrolyte at sufficient dilution is completely dissociated and, on theother hand, for sufficiently large c, cv—>0, i.e. the highly concentratedelectrolyte is dissociated only slightly. The dependence of a on c is given inFig. 1.2.

For strong electrolytes, the activity of molecules cannot be considered, asno molecules are present, and thus the concept of the dissociation constantloses its meaning. However, the experimentally determined values of thedissociation constant are finite and the values of the degree of dissociationdiffer from unity. This is not the result of incomplete dissociation, but israther connected with non-ideal behaviour (Section 1.3) and with ionassociation occurring in these solutions (see Section 1.2.4).

Arrhenius also formulated the first rational definition of acids and bases:

An acid (HA) is a substance from which hydrogen ions are dissociated insolution:

A base (BOH) is a substance splitting off hydroxide ions in solution:

This approach explained many of the properties of acids and bases andmany processes in which acids and bases appear, but not all (e.g. processes

13

in non-aqueous media, some catalytic processes, etc.). It has a drawbackcoming from the attempt to define acids and bases independently. However,as will be seen later, the acidity or basicity of substances appears only oninteraction with the medium with which they are in contact.

References

Dunsch, L., Geschichte der Elektrochemie, Deutscher Verlag furGrundstoffindustrie, Leipzig, 1985.

Ostwald, W., Die Entwicklung der Elektrochemie in gemeinverstdndlicherDarstellung, Barth, Leipzig, 1910.

1.2 Structure of Solutions

1.2.1 Classification of solvents

The classical period of electrochemistry dealt with aqueous solutions.Gradually, however, other, 'non-aqueous' solvents became important inboth chemistry and electrochemistry. For example, some important sub-stances (e.g. the Grignard reagents and other homogeneous catalysts)decompose in water. A number of important biochemical substances(proteins, enzymes, chlorophyll, vitamin B12) are insoluble in water but aresoluble, for example, in anhydrous liquid hydrogen fluoride, from whichthey can be reisolated without loss of biochemical activity. The wholealuminium industry is based on electrolysis of a solution of aluminium oxidein fused cryolite. Many more examples could be given of chemical processesemploying solvents other than water. Basically any substance can be used asa solvent at temperatures between its melting and boiling points (provided itis stable in this temperature range). Three types of solvent can bedistinguished.

Molecular solvents consist of molecules. The cohesive forces betweenneighbouring molecules in the liquid phase depend on hydrogen bonds orother 'bridges' (oxygen, halogen), on dipole-dipole interactions or on vander Waals interactions. These solvents act as dielectrics and do notappreciably conduct electric current. Autoionization occurs to a slightdegree in some of them, leading to low electric conductivity (for example2H 2O^H 3O+ + OH~; in the melt, 2HgBr2<=»HgBr++ HgBr3~; in theliquefied state, 2NO2^±NO+ + NO3").

Ionic solvents consisting of ions are mostly fused salts. However, not allsalts yield ions on melting. For example, fused HgBr2, POC13, BrF3 andothers form molecular liquids. On mixing, however, the molecular solventsH2O and H2SO4 can form ionic solvents that contain only the H3O+ andHSO4~ i°ns- Ionic solvents have high ionic electric conductivity. Most existat high temperatures (e.g., at normal pressure, NaCl between 800 and1465°C) but some salts have low melting points (e.g. ethylpyridiniumbromide at -114°C, tetramethylammonium thiocyanate at -50.5°C) and, in

14

addition, there is a number of low-melting-point eutectics (for exampleAICI3 + KC1 + NaCl in a ratio of 60:14:26 mol % melts at 94°C). The ionspresent in these solvents can be either monoatomic (for example Na+ andCl~ in fused NaCl) or polyatomic (for example cryolite—Na3AlF6—containsNa+, A1F6

3", AIF4- and F" ions).

1.2.2 Liquid structure

Molecular liquids are not at all amorphous, as would first appear.Methods of structural analysis (X-ray diffraction, NMR, IR and Ramanspectroscopy) have demonstrated that the liquid retains the structure of theoriginal crystal to a certain degree. Water is the most ordered solvent (andhas been investigated most extensively). Under normal conditions, 70 percent of the water molecules exist in 'ice floes', clusters of about 50 moleculeswith a structure similar to that of ice and a mean lifetime of 10~ns.Hydrogen bonds lead to intermolecular spatial association. Hydrogen bondsare also formed in other solvents, but result in the formation of chains (e.g.in alcohols) or rings (e.g. rings containing six molecules are formed in liquidHF). Thus, the degree of organization is lower in these solvents than inwater, although the strength of hydrogen bonds increases in the order: HC1,H2SO4 (practically monomers) < NH3 < H2O < HF. Mixing of a highlyorganized solvent with a less organized one leads to structure modification.If, for example, ethanol is added to water, ethanol molecules first enter thewater structure and strengthen it; at higher concentrations this order isreversed.

It is not the purpose of chemistry, but rather of statistical thermo-dynamics, to formulate a theory of the structure of water. Such a theoryshould be able to calculate the properties of water, especially with regard totheir dependence on temperature. So far, no theory has been formulatedwhose equations do not contain adjustable parameters (up to eight in sometheories). These include continuum and mixture theories. The continuumtheory is based on the concept of a continuous change of the parameters ofthe water molecule with temperature. Recently, however, theories based ona model of a mixture have become more popular. It is assumed that liquidwater is a mixture of structurally different species with various densities.With increasing temperature, there is a decrease in the number oflow-density species, compensated by the usual thermal expansion of liquids,leading to the formation of the well-known maximum on the temperaturedependence of the density of water (0.999973 g • cm"3 at 3.98°C).

There are various theories on the structure of these species and their size.Some authors have assumed the presence of monomers and oligomers up topentamers, with the open structure of ice I, while others deny the presenceof monomers. Other authors assume the presence of the structure of ice Iwith loosely arranged six-membered rings and of structures similar to that ofice III with tightly packed rings. Most often, it is assumed that the structure

15

consists of clusters of the ice I type, with various degrees of polymerization,with the maximum of the cluster size distribution in the region of oligomersand with a low concentration of large species.

The following simple concept is sufficient for our purposes. To a givenwater molecule, two further water molecules can become bonded fairlystrongly through their negative 'ends' by electrostatic forces in the directionof the O—H bond (hydrogen bonds). Additional two water molecules arethen bound to the original molecule through their positive 'ends'. In thesterically most favourable position these two molecules occupy the positionsof the remaining two apices of a tetrahedron whose first two apices lie in thedirection of the O—H bonds. This process results in a tetrahedral structure,the effective charge distribution in the water molecule being quadrupolar. Inthe solid state, each water molecule has four nearest neighbours (Fig. 1.3).This is a very 'open' arrangement that collapses partially on melting (thedensity of water is larger than that of ice). Thus, liquid water retains clustersof molecules with the structure of ice that constantly collapse and reform.About half of the water molecules are present in these clusters at a giveninstant. X-ray structural analysis indicates that, in the liquid state, eachwater molecule has an average of 4.5 nearest neighbours. This is far lessthan would correspond to the most closely packed arrangement (12 nearestneighbours). The existence of a certain degree of ordering in liquid watercan also explain the unusually high value of the heat of vaporization,entropy of vaporization, boiling point, and dielectric constant of watercompared with similar simple substances, such as hydrogen sulphide,hydrogen fluoride, and ammonia.

Ionic liquids are also not completely randomly arranged but have astructure similar to that of a crystal. However, in contrast to crystals, theionic liquid structure contains far more vacancies, interstitial cavities,dislocations, and other perturbations.

1.2.3 lonsolvation

If a substance is to be dissolved, its ions or molecules must first moveapart and then force their way between the solvent molecules which interactwith the solute particles. If an ionic crystal is dissolved, electrostaticinteraction forces must be overcome between the ions. The higher thedielectric constant of the solvent, the more effective this process is. Thesolvent-solute interaction is termed ion solvation {ion hydration in aqueoussolutions). The importance of this phenomenon follows from comparison ofthe energy changes accompanying solvation of ions and uncharged mole-cules: for monovalent ions, the enthalpy of hydration is about400 kJ • mol"1, and equals about 12 kJ • mol"1 for simple non-polar speciessuch as argon or methane.

The simplest theory of interactions between ions and the solvent,proposed by M. Born, assumed that the ions are spheres with a radius of rt

16

Fig. 1.3 The hexagonal array of water molecules in ice I. Even at lowtemperatures the hydrogen atoms (smaller circles) are randomly ordered

and charge zte (e is the elementary charge, i.e. the charge of a proton),located in a continuous, structureless dielectric with permittivity e. Thebalance of the changes in the Gibbs free energy during ion solvation isbased on the following hypothetical process. Consider an ion in a vacuumwhose charge zte is initially discharged, yielding electric work wx to thesystem,

• • £where ip1 is the electric potential at the surface of the ion,

.,. _ q

(1.2.1)

(1.2.2)

17

The quantity e0 is the permittivity of a vacuum (eo = 8.85418782(7) x10~"12F-m"1) and q is the charge of the ion, variable during discharging.Substitution of (1.2.2) into Eq. (1.2.1) and integration yield

fiZie

qdq= - ^ 1 . (1.2.3)8jr£r

The uncharged ion is transferred into a solvent with permittivity e = De0,where D is the relative permittivity (dielectric constant) of the medium. Nowork is gained or lost in this process. In the solvent, the ion is againrecharged to the value of the electric potential at its surface,

xp2 = ? — (1.2.4)

The corresponding electric work is

1 Cz'e (ze)2

w2 = — q dq = Kl } (1.2.5)

The transfer of one mole of ions from a vacuum into the solution isconnected with work NA(wx + w2). This work is identical with the molarGibbs energy of solvation AGS,:

= NA(w1 + w2) (1.2.6)

The Born equation for the Gibbs energy of solvation is thus

zfe2NA / i 1

Although the Born equation is a rough approximation, it is often used forcomparison of the solvation effects of various solvents. The simplificationinvolved in the Born theory is based primarily on the assumption that thepermittivity of the solvent is the same in the immediate vicinity of the ion asin the pure solvent, and the work required to compress the solvent aroundthe ion is neglected.

The ionic radii are often difficult to ascertain. Mostly, crystallographicradii rc corrected by the additive term <5, with a constant common value forcations and a different constant value for all anions, are used:

rt = rc+d (1.2.8)

Pauling's ion radii are often used, with values of 6 of 0.085 nm for cationsand 0.010 nm for anions.

As every solvent has its characteristic structure, its molecules are bondedmore or less strongly to the ions in the course of solvation. Again, solvationhas a marked effect on the structure of the surrounding solvent. Thenumber of molecules bound in this way to a single ion is termed thesolvation (hydration) number of this ion.

18

The values of the dipole-dipole and ion-dipole interaction energies arerequired for estimation of the energy conditions in liquid water and inaqueous solutions. The former is given by the sum of the coulombicinteraction energies between the individual charges of the two dipoles.Assuming that both dipoles are colinear and identical (i.e. have identicalcharge q and length /, and the dipole moment equals p = ql) and that theircentres lie a distance r apart, then the dipole-dipole interaction energy isgiven by the relationship

( +

4jze0 \ r r — I r + I

4jzeor(r( 1 2 9 )

If r»l (fulfilled even for the closest approach because the moleculardimensions are large compared with the distance between the positive andnegative charges in the dipoles), then

The expression for the ion-dipole interaction energy Uid is obtainedanalogously as the sum of the energies of coulombic interaction of an ion(charge q) with charges q' and -q' on the ends of the dipole, i.e.

„ _ 1 l-qW\.q\q'\\ q\qV „ , . „

If again r »/, then

On approaching to a distance equal to their diameters (0.276 nm), watermolecules (jU = 6.23 x 10~28 C • cm) form a quite stable entity with potentialenergy Udd= —3.32X 10~20J = 0.2eV. If a comparably large univalent ionapproaches a dipolar water molecule (\q'\ = 1.6 x 10~19C), the absoluteenergy value is almost four times larger, that is Uid= -1.18 x 10"19 J =0.7 eV. If one water molecule in liquid water is replaced in the tetrahedrallattice by an uncharged particle of the same dimensions, then the fourclosest water molecules suffice to retain the original arrangement. If the newparticle has a sufficiently large charge, e.g. positive, then the arrangementmust change. Two water molecules are bonded more strongly thanpreviously and the other two must rotate their negative 'end' towards thecation (Fig. 1.4). Thus, depending on the size and charge of the ion, theoriginal arrangement can either be retained or a new, strong structure canbe formed, or some state between these two extremes emerges. The

19

N I

Fig. 1.4 Disturbing effect of a cation (void circle)on water structure. Molecules 1 and 2 show re-

versed orientation

ordering of the liquid can be completely destroyed if the ion disturbs theoriginal arrangement without forming a new one. Negative hydrationnumbers are then obtained for some ions, as if these salts produced'depolymerization' of the water structure. This phenomenon is termedstructure breaking. The destructive effect increases with increasing ionicradius, e.g. in the order

L i + < N a + < R b + < C s + ,

Cl", NO3" < Br" < r < C1O3-

and with increasing charge, e.g. in the order

Li+<Be 2 +<Al 3 +

Figure 1.5 schematically depicts a partially distorted water structure. Aregion is formed in the immediate vicinity of the ion where the watermolecules are electrostatically bonded so strongly to the ion that they losethe ability to rotate. The value of the permittivity thus decreases sharply (to6-7 compared to a value of 78.54 at 25°C in pure water). This region istermed the primary hydration sphere. Depending on conditions, regionsfurther from the centre of the ion contain a more or less distorted waterstructure and regions even further away retain the original structure.

It should be realized that the structure breaking results in an increase inthe entropy of the system and thus in a decrease in its Gibbs energy(according to the well-known relationship at constant temperature, AG =AH — TAS). If a more complex dissolved particle contains both charged(polar) and uncharged (non-polar) groups, then this entropy factorbecomes important. The structure of the solvent remains intact at the'non-polar surface' of the particle. In order to increase the entropy of thesystem this 'surface' must be as small as possible to decrease the regionwhere the solvent structure is unbroken. This is achieved by a closeapproach of two particles with their non-polar regions or by a conformationchange resulting in a contact of non-polar regions in the molecule. These,mainly entropic, effects termed hydrophobic (in general solvophobic)

20

Fig. 1.5 A scheme of hydration: (1) cation, (2) pri-mary hydration sheath (water molecules form a tetra-hedron), (3) secondary hydration shell, (4) disorganized

water, (5) normal water

interactions are, for example, the important factors affecting the conforma-tional stability of proteins in aqueous solutions.

O. J. Samoilov proposed a statistical approach to hydration. Ions insolution affect the thermal, particularly translational, motion of the solventmolecules in the immediate vicinity of the ion. This translational motion canbe identified with the exchange of the solvent molecules in the neighbour-hood of the ion. Retardation of this exchange is thus a measure of the ionsolvation.

If the water molecule remains in the vicinity of other water molecules fora time r and in the vicinity of the ion for a time rh then the ratio XJT is ameasure of the solvation. If T , / T » 1 , the water molecule is bonded verystrongly to the ion. On the other hand, r / / r < l indicates negativesolvation—the ion breaks the solvent structure and the solvent molecules inits vicinity can more readily be exchanged with other molecules than thosein the solvent alone, where a certain degree of ordering remains. Experi-mental values of the ratio T//T for aqueous solutions using H2

18O moleculeslie in the range from tenths to unities.

Some molecular solvents (such as ammonia, aliphatic amines, hexamethyl-phosphortriamide) dissolve alkali metals; solutions with molalities of morethan lOmol-kg"1 are obtained. Ammonia complexes M(NH3)6 analogous

21

to the amminates or hydrates of salts can be crystallized from thesesolutions. The solutions are blue (absorb at 7000 cm"1, corresponding to thes->p electron transition) and paramagnetic. Paramagnetic resonance meas-urements indicate that cavities in the liquid with dimensions of about 0.3 nmcontain free electrons. Compared with aqueous salt solutions, these metalsolutions have about 105 times larger molar conductivity, indicatingelectronic conductivity. Thus, the metals in solution are dissociated to yieldsolvated cations M+ and electrons e~ (also solvated). More concentratedsolutions also contain species such as M+e~, M2 and Me". The stability ofthese solutions is low because of the formation, for example, of alkali metalamides in solutions of alkali metals in liquid ammonia.

A form of solvation also occurs in ionic solvents, but coulombicinteractions predominate. The solvent structure can be conceived asresulting from the penetration of two sublattices—cationic and anionic. Thecations of the dissolved salt are then scattered throughout the cationicsublattice and the anions in the anionic sublattice. If the dissolved salt andthe solvent have a common anion, for example, then the cations arescattered throughout the cationic sublattice and the anionic lattice remainsalmost unaffected. Let us assume that only one kind of foreign ion, a cationor an anion, is introduced into ionic solvents (while the electroneutralitycondition is maintained). If the cation of the dissolved salt has a largercharge than the solvent cation, then vacancies must be scattered throughoutthe cationic sublattice so that the distribution of electric charge remainsneutral. An example of a more complicated situation is the formation ofNa+ and A1C14~ ions in an equimolar melt of A1C13 + NaCl. If the mixturecontains a small excess of A1C13 over the stoichiometric amount, thenA12C17~ ions are also present. If Cu+ ions are then dissolved in the melt (asCuCl), they remain free and are scattered throughout the cationic sublat-tice. However, in the presence of a small excess of NaCl, the anionicsublattice contains a corresponding excess of Cl~ with which the Cu+ ionsform the CuCl2" complex which is spread throughout the anionic sublattice.

Three types of methods are used to study solvation in molecular solvents.These are primarily the methods commonly used in studying the structuresof molecules. However, optical spectroscopy (IR and Raman) yields resultsthat are difficult to interpret from the point of view of solvation and are thusnot often used to measure solvation numbers. NMR is more successful, asthe chemical shifts are chiefly affected by solvation. Measurement ofsolvation-dependent kinetic quantities is often used {electrolytic mobility,diffusion coefficients, etc). These methods supply data on the region in theimmediate vicinity of the ion, i.e. the primary solvation sphere, closelyconnected to the ion and moving together with it. By means of the thirdtype of methods some static quantities (entropy and compressibility as wellas some non-thermodynamic quantities such as the dielectric constant) aremeasured. These methods also pertain to the secondary solvatibff sphere, inwhich the solvent structure is affected by the presence of ions, but the

22

solvent molecules do not move together with the ions. These methods have,understandably, been applied most often to aqueous solutions.

The ionic mobility and diffusion coefficient are also affected by the ionhydration. The particle dimensions calculated from these values by usingStokes' law (Eq. 2.6.2) do not correspond to the ionic dimensions found, forexample, from the crystal structure, and hydration numbers can becalculated from them. In the absence of further assumptions, diffusionmeasurements again yield only the sum of the hydration numbers of thecation and the anion.

Similarly, concepts of solvation must be employed in the measurement ofequilibrium quantities to explain some anomalies, primarily the salting-outeffect. Addition of an electrolyte to an aqueous solution of a non-electrolyteresults in transfer of part of the water to the hydration sheath of the ion,decreasing the amount of 'free' solvent, and the solubility of the non-electrolyte decreases. This effect depends, however, on the electrolyteselected. In addition, the activity coefficient values (obtained, for example,by measuring the freezing point) can indicate the magnitude of hydrationnumbers. Exchange of the open structure of pure water for the morecompact structure of the hydration sheath is the cause of lowercompressibility of the electrolyte solution compared to pure water and oflower apparent volumes of the ions in solution in comparison with theireffective volumes in the crystals. Again, this method yields the overallhydration number.

The fact that the water molecules forming the hydration sheath havelimited mobility, i.e. that the solution is to certain degree ordered, results inlower values of the ionic entropies. In special cases, the ionic entropy can bemeasured (e.g. from the dependence of the standard potential on thetemperature for electrodes of the second kind). Otherwise, the heat ofsolution is the measurable quantity. Knowledge of the lattice energy thenpermits calculation of the heat of hydration. For a saturated solution, theheat of solution is equal to the product of the temperature and the entropyof solution, from which the entropy of the salt in the solution can be found.However, the absolute value of the entropy of the crystal must be obtainedfrom the dependence of its thermal capacity on the temperature down tovery low temperatures. The value of the entropy of the salt can then yieldthe overall hydration number. It is, however, difficult to separate thecontributions of the cation and of the anion.

Various methods have yielded the following average values of the primaryhydration numbers of the alkali metal cations (the number of methods usedis given in brackets and the results are rounded off to the nearest integers):Li+ 5 (5), Na+ 5 (5), K+ 4 (4), Rb+ 3 (4), and for the halide ions: F" 4 (3),Cl~ 1 (3), Br" 1 (3), I" 1 (2). The error is ±1 water molecule (except forK+, where the error is ±2). For divalent ions the values vary between 10and 14 (according to G. Kortiim). For illustration of the variability of theresults obtained by various methods, the values obtained for the Na+ ion

23

from mobility measurements were 2-4, from the entropy 4, from thecompressibility 6-7, from molar volumes 5, from diffusion 1 and fromactivity coefficients also 1. For the Cl~ ion, these methods yielded the values(in the same order): 4, 3, 0, —1, 0, 0, 1. Of the divalent ions, for example,solution of Mg2+ was measured by the mobility method, yielding a value of10 to 13, entropy of 13, compressibility of 16 and activity coefficients of 5(according to B. E. Conway and J. O'M. Bockris).

Remarkable data on primary hydration shells are obtained in non-aqueous solvents containing a definite amount of water. Thus, nitrobenzenesaturated with water contains about 0.2 M H2O. Because of much higherdipole moment of water than of nitrobenzene, the ions will be preferentiallysolvated by water. Under these conditions the following values of hydrationnumbers were obtained: Li+ 6.5, H+ 5.5, Ag+ 4.4, Na+ 3.9, K+ 1.5, Tl+ 1.0,Rb+0.8, Cs+0.5, tetraethylammonium ion 0.0, CIO4-O.4, NO^ 1.4 andtetraphenylborate anion 0.0 (assumption).

1.2.4 Ion association

As already mentioned, the criterion of complete ionization is thefulfilment of the Kohlrausch and Onsager equations (2.4.15) and (2.4.26)stating that the molar conductivity of the solution has to decrease linearlywith the square root of its concentration. However, these relationships arevalid at moderate concentrations only. At high concentrations, distinctdeviations are observed which can partly be ascribed to non-bondingelectrostatic and other interaction of more complicated nature (cf. p. 38)and partly to ionic bond formation between ions of opposite charge, i.e. toion association (ion-pair formation). The separation of these two effects isindeed rather difficult.

The species appearing as strong electrolytes in aqueous solutions lose thisproperty in low-permittivity solvents. The ion-pair formation converts themto a sort of weak electrolyte. In solvents of very low-permittivity (dioxan,benzene) even ion triplets and quadruplets are formed.

The formation of an ion pair

A+ + B"^±A+B- (1.2.13)

is described by the association constant K.ASS and the degree of association

a.dSS

Some values of association constants are listed in Table 1.1.The dependence of ion-pairing on the dielectric constant is illustrated in

the following example. The formation of K+CP ion pairs in aqueous KC1solutions has not been demonstrated. In methanol (D = 32.6), for KC1,

24

Table 1.1 Values of log K.ass for some ion pairs in aqueous solutions at 25°C(according to C. W. Davies), where — indicates that ion pair formation could

not be proved

OHF"crBr-NO3~SO4

2~s 2o 3

2P 3 O 9

3 -P2O7

4

P3O105"

I T

-0.08

—

—0.6

3.13.9

Na+

-0 .7

—

-0 .60.70.61.162.42.7

K+

—

—

-0.21.00.9

2.32.7

Ag+

2.30.43.24.4

- 0 . 21.38.8

TT

0.80.10.51.00.31.41.9

Ca2+

1.301.0

0.282.281.953.466.88.1

Cu2+

1.230.40.0

2.36

Fe3+

12.06.041.50.601.0

logKa s s = 1.15, so that at c = 0 . 1 m o l - d m 3, 32 per cent of the KC1 ispresent as K+C1~ pairs. In acetic acid (D = 6.2), log Kass = 6.9, i.e. at thesame concentration aass = 99.9 per cent. A remarkable example is theassociation of 3 x 10~5 M tetraisopentylammonium nitrate in pure water andin pure dioxan (D = 2 . 2 ) . While in water the concentration of ion pairs isnegligible, the concentration of free nitrate ions in pure dioxan is 8 x10~1 2mol • dm" 3 . Thus , the free ions are practically absent in low permit-tivity solvents (D < 5).

J. Bjerrum (1926) first developed the theory of ion association. H eintroduced the concept of a certain critical distance between the cation andthe anion at which the electrostatic attractive force is balanced by the meanforce corresponding to thermal motion. The energy of the ion is at aminimum at this distance. The method of calculation is analogous to that ofDebye and Hiickel in the theory of activity coefficients (see Section 1.3.1).The probability Pi dr has to be found for the ith ion species to be present ina volume element in the shape of a spherical shell with thickness dr at asufficiently small distance r from the central ion (index k).

When using the Boltzmann distribution, the relationship for the fractionof ions of the ith kind dNj/Ni in this spatial element is

r (1.2.15)

where z^ip is the potential energy of the ion (i .e. the electrostatic energynecessary for transferring the ion from infinite distance). For small r values,the potential ip can be expressed only in terms of the contribution of thecentral ion, ip = zke/4jzer, so that the probability density Pt is then given bythe equation

d ztzke2 \

* ) (1.2.16)

25

I

I

Distance r, nm

Fig. 1.6 Bjerrum's curves of probability: (1) unchargedparticles, (2) two ions with opposite signs (After C. W.

Davies)

whose shape is depicted (for ions k and i of opposite sign) in Fig. 1.6. Asharp minimum lies at the critical distance rmin, obtained from Eq. (1.2.16)using the condition d/^/dr = 0:

SjtekT

For aqueous solutions at 25°C, rmin = 3.057 x 10"10 ztzk (in metres), i.e.about 0.3 nm, for a uni-univalent electrolyte and about 1.4 nm for abi-bivalent electrolyte. This distance is considered to be the maximumdistance beyond which formation of ion pairs does not occur.

Integration of Eq. (1.2.15) permits calculation of the fraction of ionspresent in the associated state and thus the degree of association and theassociation constant

•fexpy <\y (1.2.18)

where y = 2rmjr and b = 2rmin/a, a being the distance of closest approachof the ions. J. T. Denison and J. B. Ramsey, W. R. Gilkerson and R. M.Fuoss obtained simplified expressions in the form

2

-In ztzkeAnkTa e AnkTae

(1.2.19)

26

Bjerrum's theory includes approximations that are not fully justified: theions are considered to be spheres, the dielectric constant in the vicinity of theion is considered to be equal to that in the pure solvent, the possibility ofinteractions between ions other than pair formation (e.g. the formation ofhydrogen bonds) is neglected and the effect of ion solvation duringformation of ion pairs is not considered (the effect of the solvation onion-pair structure is illustrated in Fig. 1.7).

Although the Bjerrum theory is thus not in general quantitativelyapplicable, the concept of ion association is very useful. It has assisted in anexplanation of various phenomena observed in the study of homogeneous

Fig. 1.7 Possible hydration modes of an ionpair: (A) contact of primary hydration shells,(B) sharing of primary hydration shells, (C)

direct contact of ions

27

catalysis, electrical mobility, the behaviour of polyelectrolytes, etc.Ion pairs (A+B~) are formed in fused salts through a process in which the

cations or anions of the solvent and of the solute exchange positions in thesolvent lattice until the cations and anions occupy neighbouring positions. Ifthe solvent is denoted as XY, then this process can be expressed by thescheme:

x+

Y~X+A+Y"

Y+ X+

Y"

X+

B~X+

Y"X+

Y~

X+

?± Y"

X+

Y"A+

Y~

XBX

+Y")"X+^

(+Y"

rx+

CY"According to M. Blander the association constant depends on the change inthe Gibbs energy, AG, during exchange of a solvent anion Y~, next to acation of the dissolved salt A+, for an anion of the dissolved salt B~according to the relationship

l n l + - = - ^ (1-2.20)

where n is the coordination number of the cation A+. The energy AG isproportional to the combination of the reciprocal values of the distancesbetween the centres of the corresponding pairs of ions (i.e. the sum of theradii of these ions):

(1.2.21)

Clearly, it is the size of ions that is decisive in ion-pair formation.Moreover, the coulombic interactions can extend even to more distantneighbours. The Blander equation is then, of course, no longer applicable.

References

Ben-Nairn, A., Hydrophobic Interactions, Plenum Press, New York, 1980.Bloom, H., and J. O'M. Bockris, Molten electrolytes, MAE, 2, 262 (1959).Burger, K., Solvation, Ionic and Complex Formation Reactions in Non-aqueous

Solvents, Elsevier, Amsterdam, 1983.Case, B., Ion solvation, Chapter 2 of Reactions of Molecules at Electrodes (Ed. N. S.

Hush), Wiley-Interscience, London, 1971.Conway, B. E., Ionic Hydration in Chemistry and Biophysics, Elsevier, Amsterdam,

1981.Davies, C. W., Ion Association, Butterworths, London, 1962.Debye, P., Polar Molecules, Dover Publication Co., New York, 1945.Denison, J. T., and J. B. Ramsey, /. Am. Chem. Soc, 11, 2615 (1955).Desnoyers, J. E., and C. Jolicoeur, Ionic solvation, CTE, 5, Chap. 1 (1982).Dogonadze, R. R., E. Kalman, A. A. Kornyshev, and J. Ulstrup (Eds), The

Chemical Physics of Solvation, Elsevier, Amsterdam, 1985.Eisenberg, D., and W. Kautzmann, The Structure and Properties of Water, Oxford

University Press, Oxford, 1969.

28

Fuoss, R. M., and F. Accascina, Electrolytic Conductance, Intersceince Publishers,New York, 1959.

Gilkerson, W. R., /. Chem. Phys., 25, 1199 (1956).Inmann, D., and D. G. Lovering (Eds), Ionic Liquids, Plenum Press, New York,

1981.Klotz, I. M., Structure of water, in Membranes and Ion Transport (Ed. E. E.

Bittar), Vol. I, John Wiley & Sons, New York, 1970.Mamantov, G. (Ed.), Characterization of Non-aqueous Solvents, Plenum Press,

New York, 1978.Marcus, Y., Ion Solvation, John Wiley & Sons, Chichester, 1986.Papatheodorou, G. W., Structure and thermodynamics of molten salts, CTE, 5,

Chap. 5 (1982).Samoilov, O. Ya., Structure of Aqueous Electrolyte Solutions and Hydration of Ions,

Consultants Bureau, New York, 1965.Skarda, V., J. Rais, and M. Kyrs, /. Nucl. Inorg. Chem., 41, 1443 (1979).Sundheim, G., Fused Salts, McGraw-Hill, New York, 1964.Tanaka, N., H. Ohtuki, and R. Tamamushi (Eds), Ions and Molecules in Solution,

Elsevier, Amsterdam, 1983.Tanford, C , The Hydrophobic Effect: Formation of Micelles and Biological

Membranes, 2nd ed., John Wiley & Sons, New York, 1980.Tremillon, B., Chemistry in Non-aqueous Solvents, Reidel, Dordrecht, 1974.

1.3 Interionic Interactions

Thermodynamics describes the behaviour of systems in terms of quan-tities and functions of state, but cannot express these quantities in terms ofmodel concepts and assumptions on the structure of the system, inter-molecular forces, etc. This is also true of the activity coefficients; thermo-dynamics defines these quantities and gives their dependence on thetemperature, pressure and composition, but cannot interpret them from thepoint of view of intermolecular interactions. Every theoretical expression ofthe activity coefficients as a function of the composition of the solution isnecessarily based on extrathermodynamic, mainly statistical concepts. Thisapproach makes it possible to elaborate quantitatively the theory ofindividual activity coefficients. Their values are of paramount importance,for example, for operational definition of the pH and its potentiometricdetermination (Section 3.3.2), for potentiometric measurement with ion-selective electrodes (Section 6.3), in general for all the systems where liquidjunctions appear (Section 2.5.3), etc.

The expression for the chemical potential of a component of a realsolution can be separated into two terms:

& -tf= AjU, = RT \nxi + RT In y,-

= A^, i d+AMf (1.3.1)

where Afiild = RT\nXi is the difference in the chemical potentials betweenthe standard and actual states, for ideal behaviour, and Apf is thecorrection for the real behaviour, which can be identified with theexpression containing the activity coefficient. It can further be written for all

29

AjU,'s that

i,f-(^f-£) (1.3.2)\ 3nt /T,p,nj*ni

Thus, the activity coefficient can be calculated if AGE is known; this is thedifference between the work of the reversible transition of the real system atconstant temperature and pressure from the standard to the actual state andthe work in the same process in the ideal system:

RT In Yi = (?^) (1.3-3)

As has already been mentioned, to carry out such a calculation is not amatter of thermodynamics, but requires adopting certain assumptions onthe structure of the system and on interactions between particles.

According to the statistical thermodynamics of solutions the logarithm ofthe activity coefficient for an electrolyte solution can be expanded in theseries

In y± = <xx" + £*i + yx\ + • • • (1.3.4)

where ny a> j3, . . . are constants (where a < 0, 0 < n < 1). These relation-ships have been verified experimentally. The term ax" describes the effectof long-range interactions in electrolyte solutions (i.e. ion-ion interactionsin contrast to short-range dipole-dipole and ion-dipole interactions; seeSection 1.2.3). At medium concentrations, the value of this term iscomparable with that of the term fixly and at low concentrations up to10~3 mol • dm"3 the term ax" predominates. The aim of the theory ofelectrolytes is to provide a theoretical interpretation of the coefficients inEq. (1.3.4). At low concentrations, the Debye-Hiickel theory is valid; thistheory neglects all types of interaction except for electrostatic, beingsatisfied with calculation of the term ax". The Debye-Hiickel theory formsthe basis and is still the valid nucleus of all the other theories of strongelectrolytes, which are more or less elaborations or modifications of theoriginal concepts.

1.3.1 The Debye-Hiickel limiting law

In infinitely dilute solutions (in the standard state) ions do not interact,their electric field corresponds to that of point charges located at very largedistances and the solution behaves ideally. As the solution becomes moreconcentrated, the ions approach one another, whence their fields becomedeformed. This process is connected with electrical work depending on theinteractions of the ions. Differentiation of this quantity with respect to nt

permits calculation of the activity coefficient; this differentiation is identicalwith the differentiation 5GE/<9n/ and thus with the term RT In y,.

30

The work connected with the mutual approach of the ions and deforma-tion of their electric fields cannot be calculated directly. However, becausethis work depends on a change in the thermodynamic functions of state, itsvalue depends only on the initial and final states, and the processes can beseparated into several steps. According to P. Debye and E. Hiickel thisseparation is carried out so that the actual concentration change, i.e. theactual approach of the ions, occurs with uncharged species. The ions aresupposed to be discharged in the standard state, i.e. they behave at theconcentration 1 mol • dm"3 as if they were at infinite dilution. Work wl

connected with this process can be readily calculated, as discharging ofisolated ions is considered. The ion concentration is then changed from thestandard state to concentration c. This transition is connected only withwork corresponding to ideal behaviour, and not with electrical work. Theaction of forces other than electrostatic interionic forces in the electrolytesolution is neglected, as these are far more short-range interactions thanelectrostatic forces. For recharging of the ions at concentration c electricalwork H>2 is expended. The calculation of work w2 is more complicated, asthis work is connected with the formation of a space charge. Debye andHiickel introduced the concept of an ionic atmosphere; i.e. an excess chargedue to ions with the opposite charge is concentrated around each ion andthe charge density is distributed in the ionic atmosphere according to theMaxwell-Boltzmann distribution law. This excess charge is equal to thecharge of the ion considered (with opposite sign).

The Debye-Huckel theory yields the coefficient yc, but the wholesubsequent calculation is accompanied by numerous approximations, validonly at high dilutions, so that in the whole region where the theory is valid itmay be assumed that yx~ ym~ yc-

First, electric work wx and w2 is calculated for a single ion species denotedas k. For wx the same procedure as for the quantity w2 in the Born treatmentof solvation Gibbs energy (Eq. (1.2.5) will be used giving (e = Ds0))

where r is the distance from the centre of the ion.At the final concentration c, the potential ipk at distance r is given not

only by the potential of the ion, ty\, but also by the potentials of thesurrounding ions. Debye and Hiickel assumed that a spherical ionicatmosphere of statistically prevailing ions with opposite charge formsaround each ion, giving rise to the potential ipa. Thus y>k = ipo

k+ \pa. Thepotential of the space charge density p is given by the Poisson equation

V2ii>= - - (1.3.6)

where V2 is the Laplace operator, whose form depends on the coordinate

31

system used. If the volume element dV contains dNt ions of sort / withcharge ztet the space charge density p, formed by ions of sort / is given as

A = ^ , (1.3.7)

If the origin of the coordinate system is located in the centre of the centralA:th ion, the number of particles dTV, in the volume dV is given by adistribution function, expressed by Debye and Hiickel in terms of theBoltzmann distribution law (cf. p. 215)

dA/i = Nt exp ( Z ^ ) dV (1.3.8)

Here, k is the Boltzmann constant and Nt = N;/V is the total number of ionsof sort i divided by the total volume. Thus

(:^) ( L 3 . 9 )

The exponential is expanded in a series and, except for unity, only thelinear term is retained (this is permissible only when the electrostatic energye \Zi\pk\ is small compared with the energy of thermal motion kT):

(Here SA^z, = 0 because of the electroneutrality condition.) Equation(1.3.10) is substituted into Eq. (1.3.6) and the Laplace operator is expressedin polar coordinates (for the spherically symmetric problem):

(The important quantity K will be discussed later.) The general integral ofEq. (1.3.11) is

where kx and k2 are integration constants determined from the boundaryconditions. If r—><*>, then t/ —>0, from which it follows that k2 = 0. Theconstant kx is determined by substitution of the expression for ipk into Eq.(1.3.10) for p. As the electroneutrality of the ion-ionic atmosphere systemis required, the integration of the space charge around the central ion overthe whole volume of the solution yields a charge of the same magnitude andof the opposite sign to that of the central ion. The volume element dV is

32

expressed in polar coordinates as dV = 4jrr2 dr. Thus,

j p dV = j Anr2p dr = - f Anr2 £ tyz2^ dr

"*rdr = -zke (1.3.13)

The Debye-Hiickel limiting law is the least accurate approximation to theactual situation, analogous to the ideal gas law. It is based on theassumption that the ions are material points and that the potential ofthe ionic atmosphere is distributed from r = 0 to r—>oo. Within these limitsthe last equation is integrated by parts yielding, for constant klf the valueezk/Afte. Potential \pk is given by the expression

^ c ( 1 3 1 4 )Aner After Afte After 4JTCLD

The final approximate form of Eq. (1.3.14) was again obtained byexpanding the exponential in a series and retaining only the linear term.Obviously, the potential tpk can be expanded to give two terms, the first ofwhich {ezklAfter) describes the contribution of the central ion and thesecond (eZk/AfteL^) the contribution of the ionic atmosphere.

The ionic atmosphere can thus be replaced by the charge at a distance ofLU = K~1 from the central ion. The quantity LD is usually termed theeffective radius of the ionic atmosphere or the Debye length. The parameterK is directly related to the ionic strength /

/ = ii)Wr/z? = i;c|.z? (1.3.15)

i=i *=i

K2 = (2e2NA/ekT)I (1.3.16)

When numerical values are assigned to the constants in Eq. (1.3.16), it gives

/ TD\( m (1.3.17)

and, for water at 25°C,

LD = 9.6223 x 1(T9/-1/2 m, or

LD = 0.30428/" 1/2nm

(The first value is valid for basic SI units, the second is more practical: / inmoles per cubic decimetre, LD in nanometres.) The radii of the ionicatmosphere for various solution concentrations of a single binary electrolyte(for which / = — \z+z_vc) are listed in Table 1.2.

33

Table 1.2 Radii of ionic atmosphere LD (in nm) at variousconcentrations of aqueous solutions at 25°C (according to Eq.

1.3.17)

c (mol • dm 3)

1(T5

io-4

io-3

1(T2

io-1

1

1-1

96.130.49.613.040.960.30

Charge type

1-2

55.517.55.551.750.560.18

of the

2-2

48.015.24.801.520.480.15

electrolyte

1-3

39.212.43.921.240.390.12

2-3

24.87.82.480.780.250.08

The work expended in recharging ion k is

0 ^ ^

The overall electrical work connected with the described process for ion k isthen

-zle2K(1.3.19)

and, for all the ions present in volume of the solution Vy

Vp2rY N 72 P

3M3/2

Work Wel is then identified with the correction for non-ideal behaviourAGE and the activity coefficient is obtained from the equation

(1.3.2!)T,niitnk

Differentiation assuming that V is independent of n, (which is fulfilled forpoint charges) yields

_ . o g v3nk

( 1 3 2 2 )

34

It is convenient to introduce

(1.3.23)In 10 x 4jzV2(£kT)3/2

yielding the usual form of the equation for the activity coefficient (cf. p. 11):

log y* =-,4z2VZ (1.3.24)

termed the Debye-Hilckel limiting law. Coefficient A has the numericalvalue

A = 5.77057 x 104(Dr)"3/2 m3/2 • moP1/2

= 1.82481 x 10\DT)-3/2 dm3/2 • mol"1/2

and, for water at 25°C,

A = 1.61039 x 10"2 m3/2 • mol"m

= 0.50925 dm3/2-mor1/2

In these expressions, the first value is valid for basic SI units and the secondfor / in moles per cubic decimetre substituted into Eq. (1.3.24). Equation(1.3.24) is a very rough approximation and does not involve the individualcharacteristics of ion k. It is valid for a uni-univalent electrolyte only up toan ionic strength of 10~3 mol • dm"3 (see also Fig. 1.8).

In view of the definition of the mean activity coefficient and of theelectroneutrality condition, v+z+= — v_z_, the limiting law also has theform

logy±=^z+z_V7 (1.3.25)

1.3.2 More rigorous Debye-Hiickel treatment of the activity coefficient

A more rigorous approach requires integration of Eq. (1.3.13) not overthe whole volume of the solution but over the effective volume after thevolume of the central ion has been excluded, as this region is not accessiblefor the ionic atmosphere. Thus, integration is carried out from r = ay i.e. fromthe distance of closest approach, equal to the effective ion diameter, whichis the smallest mean distance to which the centres of other ions can approachthe central ion. This value varies for various electrolytes (Table 1.3). In thisrefined procedure the integration constant appearing in Eq. (1.3.13) attainsthe value kx = [zkel4ne{\ + KO)\ exp {KO) and the potential \pk =zke/4jt£[r(l + Ka)]'1 exp [ic(a — r)]. This expression can again be separatedinto the contribution of the isolated central ion ip°k and the contribution of theionic atmosphere tya. As a result of the principle of linear field superposi-tion, these two quantities can be added algebraically. The contribution of the

35

Fig. 1.8 Dependence of the mean activity coefficient y±>cof NaCl on the square root of molar concentration c at25°C. Circles are experimental points. Curve 1 was calcu-lated according to the Debye-Hiickel limiting law (1.3.25),curve 2 according to the approximation aB = 1 (Eq.1.3.32); curve 3 according to the Debye-Hiickel equation(1.3.31), fl = 325nm; curve 4 according to the Bates-Guggenheim approximation (1.3.33); curve 5 according tothe Bates-Guggenheim approximation + linear term0.1 C; curve 6 according to Eq. (1.3.38) for a=0.4nm,

C = 0.055 dm5-mor1

ionic atmosphere is then given by the expressions

TK Alter \l + Ka

zke K zke

- 1 (1.3.26)

(1.3.27)

because at a distance of r < a, no ion of the ionic atmosphere can be present

36

Table 1.3 Effective ion diameters (According to B. E. Conway)

Ion a (nm)

Rb+, Cs+, NH4+, Tl+, Ag+ 0.25

K+, cr, Br, r, o r , NO2-, NO3- 0.3OH~, F" , CNS", NCO~, HS", C1O3", C1O4", BrO3~, IO4", MnO4", 0.35

HCOCT, citrate", CH3NH3+

Hg22+, SO4

2-, S2O32", S2O6

2", S2O82", Se4

2", CrO42~, HPO4

2~, 0.4PO4

3~, Fe(CN)63~, Cr(NH3)6

3+, Co(NH3)63+, Co(NH3)5H2O3+,

NH3+CH2COOH, C2H5NH3

+

N a \ CdCl+, C1O2", IO3", HCO3", H2PO4~, H S ( V , H2AsO4-, 0.45Pb2 + , CO3

2", SO32", MoO4

2", Co(NH3)5Cl2+, Fe(CN)6NO2",CH3COO", CH2C1COCT, (CH3)4N+, (C2H5)2NH2

+,NH2CH2COO"

Sr2+, Ba2+, Ra2+ , Cd2+, Hg2+ , S2~, S2O42", WO4

2", Fe(CN)64-, 0.5

CHC12COO~, CC13COO+, (C2H5)3NH+, C3H7NH3+

Li+ , Ca2+, Cu2+, Zn2+ , Sn2+, Mn2+, Fe2 + , Ni2+, Co2+ , Co(ethylendi- 0.6amine)3

3+, C6H5COO', C6H4OHCOCT, (C2H5)4N+, (C3H7)2NH2+

Mg2+, Be2+ 0.8H + , Al3+ , Fe3 + , Cr3+, Sc3+, La3+, In3+, Ce3+ , Pr3+, Nd3+, Sm3+ 0.9Th4+, Zr4+ , Ce4+ , Sn4+ 1.1

and therefore the potential remains constant and equal to the value forr = a.

In view of this equation the effect of the ionic atmosphere on thepotential of the central ion is equivalent to the effect of a charge of the samemagnitude (that is — zke) distributed over the surface of a sphere with aradius of a + LD around the central ion. In very dilute solutions, LD » a; inmore concentrated solutions, the Debye length LD is comparable to or evensmaller than a. The radius of the ionic atmosphere calculated from thecentre of the central ion is then LD + a.

Work w2 can now be calculated and, from this value, the total electricwork Wel, found in an analogous way to that previously used for the case ofthe limiting law, yielding

Ve2 Y Nz2

W*i= - A 3 V tln (! + K«) - Ka + 2(*a)] (L3.28)

(The expression in square brackets follows from the integral a3 J [q2/(l +Ka)] dq after introducing the limits 0 and zke.)

The expression for the activity coefficient is calculated from thedifferential dWjdnk (neglecting the dependence of V on nk) by substitutioninto Eq. (1.3.21). Rearrangement yields

37

where the constant A is identical with the constant in the limiting law and

/2P2N \ m

( )

The numerical value of this constant is

B = 1.5903 x 1010(D7r1/2 m1/2 • mol"1/2

= 502.90(DTym dm3/2 • mol"m • nm"1

and, for water at 25°C,

B = 1.0392 xl08m1 / 2-mor1 / 2

= 3.2864 dm3/2 • m o P m • nm"1

In the above two equations, the former value is valid for basic SI unitsand the latter value for / in moles per cubic decimetre and a in nanometres.The parameter a represents one of the difficulties connected with theDebye-Hiickel approach as its direct determination is not possible and is, inmost cases, found as an adjustable parameter for the best fit of experimentaldata in the Eq. (1.3.29). For common ions the values of effective ion radiivary from 0.3 to 0.5. Analogous to the limiting law, the mean activitycoefficient can be expressed by the equation

where a is the average of the effective ion diameters of the cation a+ and forthe anion a_ (this averaging is the source of difficulties in cases of morecomplicated systems like electrolyte mixtures). Equation (3.1.31) is adefinite improvement in comparison of the limiting law (Fig. 1.8). As theconstant B in the units mol~3/2 • nm"1 is about 3.3 the product aB being notfar from unity some authors employ a simplified equation (for aqueoussolutions at 25°C)

_Az+z_Vl

The Bates-Guggenheim equation

is also often used. Like the limiting law, the latter two equations do notrequire any specific information on the ions considered; nonetheless,they describe the y — I dependence more closely than the limiting law (seeFig. 1.8).

If the ionic strength in concentrated aqueous or non-aqueous solutions isexpressed in terms of the molalities, then the constants A' = A\fp andB1 = BVp are used, where p is the density of the solvent.

38

1.3.3 The osmotic coefficient

The osmotic pressure of an electrolyte solution n can be considered as theideal osmotic pressure JZ* decreased by the pressure jrel resulting fromelectric cohesion between ions. The work connected with a change in theconcentration of the solution is n dV = n* dV — JtcX dV. The electric part ofthis work is then jre, dV = dWeU and thus JZCX = (dWel/dV)Ttn. The osmoticcoefficient (j> is given by the ratio jt/jt*, from which it follows that

jrcl (dWc( L 3 ' 3 4 )

as the ideal osmotic pressure is jt* = kNAT £ ch In order to obtain(dWJdV)T,n> Eq. (1.3.20) is differentiated, resulting in

Consider a dilute solution of a single electrolyte, so that £ cz = c+ 4- c_,c+z+ + c_z_=0 and /=—5Z+z_c. Comparison of Eq. (1.3.35) with Eq.(1.3.22) written for y± and with Eq. (1.3.23) yields the limiting relationship

p (1.3.36)

For water at 0°C (the osmotic coefficient is mostly determined fromcryoscopic measurements),

1 _ ^ = -0.2658z+z_(vc)1/2 (1.3.37)

1.3.4 Advanced theory of activity coefficients of electrolytes

According to Eq. (1.3.31) log y should be a monotonous function of y/lwhich obviously is not the case. A possible remedy of this situation wouldbe, for example, the introduction of an additional linear term on theright-hand side of that equation,

In this equation, the coefficient C as well as the coefficient a must be takenas adjustable parameters found experimentally. The result is shown in Fig.1.8, curve 6. Otherwise a quite good approximation for ionic strengths up to0.3 is achieved by putting C = — 0.lz+z_.

Thus, a suitable refinement of the Debye-Hiickel theory must providea theoretical interpretation of the term CI. Originally this term wasqualitatively interpreted as a salting-out effect: during solvation the ions

39

orient solvent molecules so that the electrostatic forces leading to non-ideality of the solution are weakened. The greater the concentration of ions,the greater the portion of the solvent molecules that is oriented; the activitycoefficients again increase with increasing ionic strength.

Most authors have accounted for the mutual influence of ions and solventmolecules only by assuming a firmly bound solvent sheath around the ion.The structure of the bulk solvent and the influence of electrolyte concentra-tion on this structure are not taken into consideration.

The necessity of considering the solvent sheath around the ions followsfrom the fact that the effective ionic diameters a calculated from theDebye-Hiickel equation (1.3.31) using the experimental activity coefficientvalues are too large compared with the corresponding crystallographicquantities. This led to the concept that Eq. (1.3.31) is valid for the activitycoefficients of the solvated ions that form a mixture with the remaining'free' solvent differing from ideal behaviour only in the electrostaticinteraction between the solvated ions. If, on the other hand, the activitycoefficients are calculated from the experimental data, then the molality ormole fraction is expressed in terms of the amount of anhydrous electrolyteand all the solvent is considered as 'free'. This calculation yields a formalactivity coefficient for the unsolvated ions, whose existence is fictitious.Thus, if the theoretical expressions for the activity coefficients are to becompared with the experimental values, the values for the solvated speciesmust be recalculated to those for unsolvated species. This recalculation isparticularly important for concentrated solutions, where the decrease in theamount of 'free' solvent is significant.

The method of calculating these values was proposed by R. A. Robinsonand R. H. Stokes. The Gibbs energy G of a system containing 1 mole ofuni-univalent electrolyte {nx = 1, total number of moles of ions being 2) andn0 moles of solvent is expressed for solvated ions (dashed quantities) on onehand and for non-solvated ions (without dash) on the other; it is assumedthat JJ,0 = A*o and that the hydration number h (number of moles of thesolvent bound to 1 mole of the electrolyte) is independent of concentration:

+M+ + M- (1-3.39)

Chemical potentials are split up into standard and variable terms,

(*i° + ii°; + ju°_ + n°-)/RT + h^/RT + h Ina0 + 2In

+ In y+tX + In Y-,x = In y+ + In y'- (1-3.40)

For Ai0—»°°, all the logarithmic terms approach zero. Thus the combinationof the standard potentials given by the first terms in Eq. (1.3.40) equals

40

zero, so that the mean activity coefficient is given by the equation

2In y±tX = 2In y'±tX-h\nao-2\nn° + 2 h (1.3.41)n + 2

Robinson and Stokes identified the quantity 0.4343 In y±>JC with the expres-sion on the left-hand side in Eq. (1.3.31). For the system considered here,m=nl/n0M0 (where Mo is the molar mass of the solvent, for waterMo = 0.018 kg/mol), so that

In [(/i0 + 2 - h)/(n0 + 2)] = In [(1 + 2Mom - hMom)/(l + 2Mom)]

and, according to Appendix A, y±tX = y±>w(l + 2Mom). Hence,

log 7±,w = - 7 ^ ^ 7 7 - ^ / 2 log «0-log [1 +(2-/z)Mom] (1.3.42)

As in view of Eq. (1.1.18) Ina0 = -(f>mMom and l»(2-h)Mom Eq.(1.3.42) gives

log y±,m = DH + 0A343[<t)mMoh + (2 - h)M0]m (1.3.43)

where DH denotes the Debye-Hiickel term (1.3.31) while the second termon the right-hand side corresponds to CI of Eq. (1.3.38).

The Robinson-Stokes equation (1.3.43) can be used to calculate theparameters a and h from the measured values of y± m and 0W. In this waythe hydration number can also be found, but only as the sum of the ionhydration numbers of all the ions in the electrolyte 'molecule'. Theappropriate calculations have been carried out for a number of systems (seeTable 1.4). Equation (1.3.43) describes the experimental data up tomolalities of the order of units, so that it is formally very successful. Ionicdiameters vary from 0.35 to 0.62 nm. The calculated hydration numbers forsalts with a common anion retain the order expected for the cations of thesesalts (H+ > Li+ > Na+ > K+ > Rb+), but do not retain the order expected forthe anions with a common cation (I~ > Br~ > Cl~). Furthermore, these num-bers do not preserve additivity: for example /iNaci ~ ^KCI = 1.6, but /*Naci ~hKl = 3.0. If it is assumed that chloride ions are practically not hydrated andthe whole hydration number of chloride salts is attributed to the cation,then the numbers obtained are too large (e.g. 12 for CaCl2 and 13.7 forMgCl2). If the radius of the solvated cation is estimated considering thevolume of bound solvent and this is added to the crystallographic radius ofthe anion, numbers are obtained that exceed the suitable effective diametersto a differing degree, depending on the type of electrolyte (e.g. by 0.07 nmfor uni-univalent and by 0.13 nm for uni-divalent electrolytes, etc.). Thisdiscrepancy is explained by penetration of the anion solvation sheath intothe sheath of the cation (Fig. 1.7).

The contribution of short-range forces to the activity coefficient can bedescribed much better and in greater detail by the methods of the statisticalthermodynamics of liquids, which has already created several models ofelectrolyte solutions. However, the procedures employed in the statistical

41

Table 1.4 Values of the parameters a and h (Eq. 1.3.42). (According toR. A. Robinson and R. H. Stokes)

HC1HBrHIHC1O4

LiClLiBrLilL1CIO4NaClNaBrNalNaClO4

KC1KBrKINH4C1RbClRbBr

h

8.08.6

10.67.47.17.69.08.73.54.25.52.11.92.12.51.61.20.9

a (nm)

0.4470.5180.5690.5090.4320.4560.5600.5630.3970.4240.4470.4040.3630.3850.4160.3750.3490.348

RblMgCl2

MgBr2

Mgl2

CaCl2

CaBr2

Cal2

SrCl2

SrBr2

Srl2

BaCl2

BaBr2

Bal2

MnCl2

FeCl2

CoCl2

NiCl2

Zn(QO4)2

h

0.613.717.019.012.014.617.010.712.715.57.7

10.715.011.012.013.013.020.0

a (nm)

0.3560.5020.5460.6180.4730.5020.5690.4610.4890.5580.4450.4680.5440.4740.4800.4810.4860.618

thermodynamics of liquid mixtures are conceptually and mathematicallyrather complicated so that the reader must be referred to special mono-graphs on the subject. On the other hand, it matters little at this stage thatstatistical thermodynamic procedures are so far lacking in the presenttextbook. The derived equations have not yet even been worked out to anexperimentally verifiable form. In fact, the theory of ionic solutions shouldyield an equation permitting calculation of the activity coefficients of ions onthe basis of knowledge of the structure of the ion considered and of all theother components of the solution. The methods of statistical thermo-dynamics that would, in principle, be capable of carrying out this calculationare developing rapidly and it seems that the future looks promising in thisarea.

1.3.5 Mixtures of strong electrolytes

The derivation of the equations of the Debye-Huckel theory did notrequire differentiation between a solution of a single electrolyte and anelectrolyte mixture provided that the limiting law approximation Eq.(1.3.24), was used, which does not contain any specific ionic parameter. If,however, approximation (1.3.29) is to be used, containing the effective ionicdiameter a> it must be recalled that this quantity was introduced as theminimal mean distance of approach of both positive and negative ions to thecentral ion. Thus, this quantity a is in a certain sense an average of effects ofall the ions but, at the same time, a characteristic value for the given centralion.

42

If the validity of Eq. (1.3.31) is assumed for the mean activity coefficientof a given electrolyte even in a mixture of electrolytes, and quantity a iscalculated for the same measured electrolyte in various mixtures, thendifferent values are, in fact, obtained which differ for a single total solutionmolality depending on the relative representation and individual propertiesof the ionic components.

A number of authors have suggested various mixing rules, according towhich the quantity a could be calculated for a measured electrolyte in amixture, starting from the known individual parameters of the singleelectrolytes and the known composition of the solution. However, none ofthe proposed mixing relationships has found broad application. Thus, thequestion about the dependence of the mean activity coefficients of theindividual electrolytes on the relative contents of the various electrolyticcomponents was solved in a different way.

The approach introduced by E. A. Guggenheim and employed by H. S.Harned, G. Akerlof, and other authors, especially for a mixture of twoelectrolytes, is based on the Br0nsted assumption of specific ion inter-actions: in a dilute solution of two electrolytes with constant overallconcentration, the interaction between ions with charges of the same sign isnon-specific for the type of ion, while interaction between ions withopposite charges is specific.

Guggenheim used this assumption to employ Eq. (1.3.38) for the activitycoefficient of the electrolyte, where the product aB was set equal to unityand the specific interaction between oppositely charged ions was accountedfor in the term CI. Consider a mixture of two uni-univalent electrolytesAlBl and AnBn with overall molality m and individual representationsyY = mjm and yu = mu/m, where mx and mu are molalities of individualelectrolytes. According to Guggenheim,

In y, = -A* + [2y,6u + (ftIU + 6UI)(1 - Vl)]m ^

In Yn = -A* + [2(1 -y)6 + (6 + K)y]m

where the fo's are specific interaction constants (blu is the parameter of theinteraction of Ax with Bu, bu is the parameter of the interaction of Ax withBl9 etc.) and>4* = lnlO>lV7/(l + V7).

Two limiting activity coefficients can be obtained from Eq. (1.3.44) foreach electrolyte:

lim In y, = In y? = -A* 4- (bu x + bx n)mo

lim In y, = In y\ = -A* + 2bllm (1.3.45)

lim In yu = In yn = -A* + (bu { 4- bY n)m

lim In Yn = m Yu = ~^* + 2^n nm

43

Thus, lny? = lnyn .Coefficients or, and au are introduced, describing the combination of the

interaction constants:

m! o (1-3.46)

In 10 «„ = 2bUM - V , - 6,.n = '" y " " '" y "m

Combination of Eqs (1.3.44) to (1.3.46), introduction of molalities mY andmn and conversion to decadic logarithms yield the equations

log yY = log y? + alml = log y\ - <xxmYl ^

log Yu = log y?! + aumn = log y}T - auml

These relationships are termed the Harned rules and have been verifiedexperimentally up to high overall molality values (e.g. for a mixture of HC1and KC1 up to 2mol-kg~1). If this linear relationship between thelogarithm of the activity coefficient of one electrolyte and the molality of thesecond electrolyte in a mixture with constant overall molality is not fulfilled,then a further term is added, including the square of the appropriatemolality:

log Y\ = log y,1 - ari/wn $xm\x ^

log yu = log Yu ~ ocuml - /3nm?

Robinson and Stokes have demonstrated that, for simplifying assumptions,the following relationship is valid between ar's and )3's:

(a ,+ aII) = fc-2rn(i8I + i8II) (1.3.49)

where A: is a constant independent of the molality.A particular case of electrolyte mixtures occurs if one electrolyte is

present in a large excess over the others, thus determining the value of theionic strength. In this case the ionic atmospheres of all the ions are formedalmost exclusively from these excess ions. Under these conditions, theactivities of all the ions present in the solution are proportional to theirconcentrations, the activity coefficient being a function of the concentrationof the excess electrolyte alone.

The excess electrolyte is often termed the indifferent electrolyte. From thepractical point of view, solutions containing an indifferent electrolyte arevery often used in miscellaneous investigations. For example, when deter-mining equilibrium constants (e.g. apparent dissociation constants, Eq.1.1.26) it is necessary only to indicate the indifferent electrolyte and itsconcentration, as they do not change when the concentrations of thereactants are changed. Moreover, the indifferent electrolyte is important inthe study of diffusion transport (Section 2.5), for elimination of liquid

44

junction potentials (Section 2.5.3) and in the investigations of electrochemi-cal kinetics (Section 5.4).

1.3.6 Methods of measuring activity coefficients

Two types of methods are used to measure activity coefficients.Potentiometric methods that measure the mean activity coefficient of thedissolved electrolyte directly will be described in Section 3.3.3. However, ingalvanic cells with liquid junctions the electrodes respond to individual ionactivities (Section 3.2). This is particularly true for pH measurement(Sections 3.3.2 and 6.3). In these cases, extrathermodynamical proceduresdefining individual ion activities must be employed.

Thermodynamic methods also measure the activity coefficient of thesolvent (it should be recalled that the activity coefficient of the solvent isdirectly related to the osmotic coefficient—Eq. 1.1.19). As the activities ofthe components of a solution are related by the Gibbs-Duhem equation,the measured activity coefficient of the solvent can readily be used tocalculate the activity coefficient of a dissolved electrolyte.

All methods based on osmotic phenomena lead directly to measurementof activities; these include methods measuring the decrease in vapourpressure above the solution, boiling point elevation, freezing point suppres-sion and direct measurement of the osmotic pressure. Of these methods, themost commonly used is the cryoscopic method; ebulioscopic and osmometricmethods are technically more complicated and are used less frequently.Tensiometric methods are also often used, involving measurement of thevapour pressure of the solvent above the pure solvent /?£ and above thesolution p0. The ratio of these two values, po/poy gives the activity of thesolvent in the solution. Various methods are used to measure vapourpressure. Special tensiometers can be used to measure vapour pressuredirectly in tensiometric measurements. Dynamic methods involve passing aninert gas first through the pure solvent, then through a drying adsorbent,further through the solution and then through another adsorbent; the ratioof the amounts of solvent adsorbed on the two adsorbents equals the ratioof the two vapour pressures. The isopiestic method involves placing twosolutions of various electrolytes B and C in a single solvent in a special celland bringing them into equilibrium through the gaseous phase. At osmoticequilibrium, the two solutions have the same vapour pressure and the samesolvent activity; the molalities of the two solutions are measured atequilibrium, raB and mc. If, for each value of the molality of one of the twoelectrolytes, for example B, the concentration dependence of the osmoticcoefficient <j)B is also known, then <pc can be calculated from the equation

Methodically, there is no great difference between measuring the meanactivity coefficient in a solution of one electrolyte and measuring thisquantity in a mixture of electrolytes. Binary mixtures have been studiedmost extensively. If osmotic methods are used, then the coefficients ocx and

45

au can be calculated and, from these values, the activity coefficients usingthe above relationships. If the Harned rule is not valid, then similar, morecomplicated relationships are used that also contain the coefficients jSj andj8n. If potentiometric methods are used, then the activity coefficients areobtained directly, i.e. values of yY and yn for the given total molality andrelative abundances of the two electrolytes.

ReferencesConway, B. E., Ionic interactions and activity behaviour, CTE, 5, Chap. 2 (1982).Friedman, H. L., Ionic Solution Theory, Wiley-Interscience, New York, 1962.Harned, H. S., and B. B. Owen, The Physical Chemistry of Electrolytic Solutions,

Reinhold, New York, 1950.Robinson, R. A., and R. H. Stokes, Electrolyte Solutions, Butterworths, London,

1959.

1.4 Acids and Bases

The Arrhenius concept was of basic importance because it permittedquantitative treatment of a number of acid-base processes in aqueoussolutions, i.e. the behaviour of acids, bases, their salts and mixtures of thesesubstances in aqueous solutions. Nonetheless, when more experimentalmaterial was collected, particularly on reaction rates of acid-base catalysedprocesses, an increasing number of facts was found that was not clearlyinterpretable on the basis of the Arrhenius theory (e.g. in anhydrousacetone NH3 reacts with acids in the absence of OH~ and without theformation of water). It gradually became clear that a more general theorywas needed. Such a theory was developed in 1923 by J. N. Br0nsted and,independently, by T. M. Lowry.

1.4.1 DefinitionsThe generalization was based on the introduction of the concept of

donor-acceptor pairs into the theory of acids and bases; this is afundamental concept in the general interpretation of chemical reactivity. Inthe same way as a redox reaction depends on the exchange of electronsbetween the two species forming the redox system, reactions in anacid-base system also depend on the exchange of a chemically simplespecies—hydrogen cations, i.e. protons. Such a reaction is thus termedprotolytic. This approach leads to the following definitions:

An acid HA is a substance capable of splitting off a proton:

A base B is a substance capable of bonding a proton:

B + H+->BH+

Substances capable of splitting off a proton, i.e. proton donors, are termedprotogenic and, on the other hand, substances accepting protons, i.e. proton

46

accceptors, are protophilic. Some substances can exhibit both of theseproperties, i.e. can either accept or donate a proton, and are termedamphiprotic. The pairs HA — A and B — HB differ only by the presence orabsence of a proton (in the same way as the oxidized form differs from thereduced form in a redox system only by the presence or absence of anelectron). The substances HA and A~ or B and BH+ form a conjugate pair.A strong acid is conjugate with a weak base and a strong base is conjugatewith a weak acid.

An important characteristic of the definitions should be noted: nomention is made of the charge of species HA, A, B and HB. Acids andbases can be either electrically neutral molecules or ions.

As free proton cannot exist alone in solution, reactions in which a protonis split off from an acid to form a conjugate base cannot occur in anisolated system (in a homogeneous solution although this ispossible in electrolysis (Section 5.7.1)). The homogeneous solution mustcontain another base Bn that accepts a proton from the acid HAr (acid HAX

is, of course, not conjugate with base Bn). It will be seen that this secondbase can even be the solvent molecule, provided it has protophilic properties.Acid-base reactions thus depend on the exchange of a proton between an acidand a base that are not mutually conjugate^

HAl + B n ^ Ax + HBn (1.4.1)

This scheme describes, in a uniform manner, all the reactions consideredin the classical (Arrhenius) theory as different processes, giving themdifferent names. They include, for example,

HC1 + NH3-> CP + NH4+ Salt formation

HC1 + H2O-> C\~ + H3O+ Acid dissociation

H2O + N H 3 ^ OH" + NH4+ Base dissociation

H2O + H2O -> OH" + H3O+ Autoionization

H3O+ + OH" -* H2O + H2O Neutralization

NH4+ + H2O -> NH3 + H3O+ Hydrolysis

H2O + CH3COO~ -> OH~ + CH3COOH Hydrolysis

1.4.2 Solvents and self-ionization

To begin with, molecular solvents with high permittivities will beconsidered. Classification of solvents on the basis of their permittivitiesagrees roughly with classification as polar and non-polar, and the borderlinebetween these two categories is usually considered to be a relative dielectricconstant of 30-40. Below this value ion pairs are markedly formed. From

f As the acid-base equilibrium is dynamic the reaction HA + A*~ <=* A~ + HA*, where HAand HA* are identical, unceasingly takes place at a high rate but has no influence on theequilibrium properties of the system (cf. p. 99).

47

the point of view of acid-base properties, solvents can be classified asprotic—containing a dissociable proton—and aprotic—not able to dissociatea proton. This classification roughly corresponds to division on the basis ofwhether the solvent molecules are or are not bonded by hydrogen bonds inthe liquid state.

Most protic solvents have both protogenic and protophilic character, i.e.they can split off as well as bind protons. They are called, therefore,amphiprotic. These include: water, alcohols, acids (especially carboxylic),ammonia, dimethylsulphoxide and acetonitrile. Solvents that are protogenicand have weak or practically negligible protophilic character include acidsolvents, such as sulphuric acid, hydrogen fluoride, hydrogen cyanide, andformic acid.

Aprotic solvents are not protogenic, but can be protophilic, e.g. acetone,1,4-dioxan, tetrahydrof uran, dimethy Iformamide, hexamethylphos-phortriamide, propylene carbonate and sulpholane. Solvents that do notparticipate in protolytic reactions, i.e. do not donate or accept a proton, areusually chemically inert, such as benzene, chlorobenzene, chloroform,tetrachloromethane, etc.

The molecules of amphiprotic solvents which are the most important willbe designated as SH. Self-ionization occurs to a small degree in thesesolvents according to the equation

2HS^H 2S+ + S" (1.4.2)

This equilibrium is characterized by the ion product of the solvent—thermodynamic

( ) ( )or apparent (1.4.3)

The activity of the solvent molecule HS in a single-component solvent isconstant and is included in ^Hs- The concentration of ions is mostly quitelow. For example, self-ionization occurs in water according to the equation2H2O-^H3O+ + OH~. The conductivity of pure water at 18°C is only3.8 x 10~8Q"1 cm"1, yielding a degree of self-ionization of 1.4xlO~19.Thus, one H3O+ or OH~ ion is present for every 7.2 x 108 molecules ofwater. Some values of Kus

a r e listed in Table 1.5 and the temperaturedependence of the ion product of water Kw is given in Table 1.6.

The H2S+ ion is generally termed a lyonium ion and the S~ ion is termed

a lyate ion. The symbol H2S+ (for example H3O+, CH3COOH2

+, etc.)refers only to a 'proton solvated by a suitable solvent' and does not expresseither the degree of solvation (solvation number) or the structure. Forexample, two water molecules form the lyonium ion H3O+, termed theoxonium (formerly hydronium or hydroxonium) ion, and the lyate ionOH", termed the hydroxide ion.

The existence of the species, H3O+, was substantiated as early as in 1924

48

Table 1.5 Self-ionization constants of solvents. (According to B. Tremillon)

Solvent

AcetamideAcetonitrileAcetic acidHydrofluoric acidFormic acidHydrogen sulphideSulphuric acidAmmonia

DimethylsulphoxideWaterEthanolEthanolamineFormamideHydrazineMethanol

Cation

CH3CONH3+

CH3CNH+

CH3CO2H2+

H2F+

HCO2H2+

H3S2+

H3SO4+

NH4+

C2H6SOH+

H3O+

C2H5OH2+

+H3NC2H4OHHCONH3

+

N2H,+

CH3OH2+

Anion

CH3CONHCH2CNCH3CO2FHCO2~HS~HSO4

HN2~

QH 5 SO-HOC2H5O~H2NC2H4O-HCONH"N2H3"CH3O

P^HS

14.619.614.910.76.2

32.62.9

27.73233.314.019.05.5

16.81316.7

°C

10025250

25- 7 8

2525

- 6 025252525252525

by M. Volmer by the isomorphism of the monohydrate, HC1O4H2O, withNH4CIO4. The species present in the crystal obviously was H3O

+-C1O4~. Anexact proof follows from the NMR spectra: the monohydrates of themolecules of nitric, perchloric and sulphuric acids contain protons placed inthe apices of equilateral triangles while other structures would give quitedifferent spectra. The H3O+ structure is hydrated by further three watermolecules, so that the most stable species are the H9O4

+ and H7O4~ ions:

VHH

AH H

0

H

•9:H H

.H H.H - O 'H

However, the symbols H3O+ (often simply H+) and OH will be retained inthis book.

Simple self-ionization cannot be assumed a priori. The situation issometimes more complicated, e.g.

3HClO4-> C12O7 + H3O+ + C1O4"

In mixed solvents, the overall ion product is defined. For example, for a

49

Table 1.6 Ion product of water at differenttemperatures. (According to B. E. Conway)

<(°Q

05

1015202530

P*w

14.943514.733814.534614.346314.166913.996513.8330

'(°C)

354045505560

P*w

13.680113.534813.396013.261713.136913.0171

binary mixture of solvent HS with water, this product is

KHS.W = ([H2S+] + [H3O

+])([S-] + [OH"])

and the equilibria

HS + OH

HS + H3O+

2H2O

2HS

?±S" + H

?±H2S+4

^ H 3 O + -

?±H2S+4

2oH2O

HOH-

• S ~

(I)

(II)(III)

(IV)

(1.4.4)

(1.4.5)

are established. The constant of the first reaction KY indicates whether wateris a stronger acid than the solvent HS, constant Ku indicates whether wateris a stronger base than solvent HS and constants Km and Klw indicate thedegree of self-ionization of the two liquids in the mixture. The ratios of theconcentrations of the ions

[S-] , [HS] [H2S+]^ [HS][OH"] '[H2O]' [H3O

+] "[H2O](1.4.6)

thus depend on the relative strengths of HS and water as acids and basesand on the composition of the mixture. Various situations can occur in thissystem. If the acid-base properties of water and HS are roughly identical,then definition (1.4.4) is valid; however, because the concentrations of themolecules H2O and HS are given by the composition of the mixture,the dependence of ^H S w on the composition exhibits a maximum. If, onthe other hand, solvent HS is almost inert, Kx and Ku are small, theconcentrations of H2S+ and S" can be neglected in Eq. (1.4.4) and theexpression becomes that for Kw of pure water. However, the concentrationof water molecules appearing in the equation for Kw decreases withincreasing concentration of the solvent HS (which acts as an inert diluent)so that Kw decreases. If water is a sufficiently stronger acid than HS but HSis a stronger base than water, then K1<1 while Kn>l, [H3O

+] can beneglected compared to [H2S

+] and [S~] can be neglected compared to

50

[OH"], so that #HS,W = [H2S

+][OH~]. This situation occurs for a mixture ofhydrazine and water, where the equilibrium H2O + N2H4^±OH~ + N2H5

+

is established. Other combinations of acid and base strengths can bediscussed in similar terms.

1.4.3 Solutions of acids and bases

In amphiprotic solvents protolysis of acids and bases takes place. Thisreaction is also termed solvolysis, classically dissociation. The degree ofsolvolysis depends on the nature of both the dissolved acid or base and onthe solvent.

The acid HA reacts with the solvent HS according to the reaction

HA + HS<F± A" + H2S+, (1-4.7)

with

_q(A-)fl(H2S+) [A-][H2S+]KA~ fl(HA) ' * A " [HA] ( 1 A 8 )

for aqueous solutions

_q(A>(H 3+O) [A-][H3O+]

* A ~ «[HA] ' * A ~ [HA] ( 1 A 9 )

where KA is termed the true (thermodynamic) dissociation constant whileKA apparent dissociation constant of the acid. This constant gives thestrength of the acid in the given solvent: the larger the value of KA or K'A(the smaller pKA and pKA), the stronger the acid. For the degree ofdissociation a [Eq. (1.1.31)] directly applies when setting [B+] =[H2S

+]([H3O+] in aqueous solutions).

The generally accepted measure of the acidity of any solution is thelogarithm of the activity of solvated protons times —1,

pHHS= -loga(H2S+)~ -log[H2S+] (1.4.10)

giving for aqueous solutions the important definition

pH=-logfl(H3O+)-log[H3O+] (1.4.11)

The subsequent text will be restricted to aqueous solutions and con-centrations will be used instead of activities. Thus, the neutral point is givenby the condition [H2O

+] = [OH"] = K'wm,

pH = ^p/C; (1.4.12)

A weak dibasic acid H2A dissociates in two steps:

H2A 4- H2O *± HA~ + H3O+

2 +H3O+ ( • • )

51

with dissociation constants K[ and K2 and degrees of dissociation ax andfor which

[H2A] 1 - a,(1.4.14)

-][HO+] (l + )tJA][H3O]2 [HA~] 1 - a2

(simple consideration reveals that [H2A] = c(l - ax)y [HA"] = cax{\ - a2),[A2~] = ca1a2, [H3O

+] = ca1 + cala2, where c = [H2A] + [HA"] + [A2"]).For a weak acid (K[> K2« 1) and not too dilute solution (aly a2« 1), thelast pair of equations simplifies to yield K\ ~ ca\ and K2 « cocxoc2y so that

[H3O+] = cocx + cocxa2 - (K[c)m + tf2 (1.4.15)

In most systems, K2< K\. Otherwise (e.g. the first two dissociation steps forethylenediaminetetraacetic acid—EDTA) the first hydrogen ion capable ofdissociation is stabilized by the presence of the second one and candissociate only when the second begins to dissociate. As a result bothhydrogen ions dissociate in one step.

As the second dissociation constant is often much smaller than the first,a2« ocx « 1 and [H2S

+] ~cocx = (K[c)m. Thus, the acid can be consideredas monobasic with the dissociation constant K[.

Solvolysis of base B

+ S~ (1.4.16)

with a dissociation constant

is usually formulated as dissociation of the cation of the base HB+ so thatEqs (1.4.8) and (1.4.9) apply for BH+ = HA.

Obviously, for a conjugated acid-base pair

K'A(BH+)K{i = Ksu, or PK^ + pK^ = pKSH (1.4.18)

(for aqueous solutions at 25°C pA A + P # B = 14).The above treatment of moderately dilute acids and dibasic acids can be

used for analogous cases of bases. Table 1.7 lists examples of dissociationconstants of bases in aqueous solutions.

The concentration of lyonium or lyate ions cannot be increased ar-bitrarily. A reasonable limit for a 'dilute solution' can be considered to be amolality of about 1 mol kg"1 (e.g. in water, one mole of solute per 55 molesof water). When the ratio is greater than 1:55, it is difficult from athermodynamic point of view to consider the system as a solution, but itshould rather be viewed as a mixture. This situation becomes increasingly

52

Table 1.7 Dissociation constants of weak acids and bases at 25°C. (FromCRC Handbook of Chemistry and Physics)

Acid

Acetic acidBenzoic acidBoric acid (20°C)Chloroacetic acidFormic acid (20°C)Phenol (20°C)6>-Phosphoric acid

(first step)o-Phosphoric acid

(second step)o-Phosphoric acid

(third step)

p/CA(HA)

4.754.199.142.853.759.892.12

7.21

12.67

Base

AcetamideAmmoniaAnilineDimethylamineHydrazineImidazolMethylaminePyridineTrimethylamine

pKA(BH+)

0.634.754.63

10.735.776.95

10.665.259.81

less favourable when the molecular weight of the solvent increases. Therange of reasonably useful values of pHHs in a given solvent is limited onthe acid side by the value pHHs = 0 ([H2S

+]) = 1) and on the basic side bythe value pHHs = P^Hs (when [S~] = 1); the neutral point then lies in themiddle where pHHs = 2P^HS-

The strongest acid that can be introduced into solution is the H2S+ ionand the strongest base is the S~ ion. However, these ions can be added tothe solution only in the form of a molecule. There are three mainpossibilities. If a strong acid is dissolved in solution, it quantitatively yieldsprotons that are immediately solvated to form H2S+ ions. The correspond-ing conjugate base cannot recombine with protons in a given solvent (e.g.introduction of HC1 into solution yields Cl~ ions) since it has no basicproperties in this solvent. The value of pHHS is given directly by theconcentration of the dissolved strong acid cA according to the simplerelationship pHHS= ~logcA. If a strong base is dissolved whose conjugateacid form cannot dissociate under the given conditions as bases such asKOH in water or CH3COOK in glacial acetic acid (the K+ cation has nodissociable hydrogen), the pHHs value is again given by the concentration ofthe dissolved strong base, in this case according to the relationship

After discussing the general properties of an amphiprotic solvent severalexamples will now be considered. As the Arrhenius concept took onlyaqueous solutions into account it became a seemingly unambiguous basis forthe classification of individual chemical substances as strong or weak acidsor bases. However, the designation of a given substance as an acid or base,weak or strong, lacks a logical foundation. Of decisive importance is thebehaviour of a given substance with respect to a given solvent, where it canact as a strong or weak acid or base. This generalization follows from theconcepts of Br0nsted and Lowry.

53

Water. The pH range that can reasonably be used extends from 0 to 14,with the neutral point at pH 7. A solution with pH<7 is acidic and withpH > 7 is basic. The commonest strong acids are HC1O4, H2SO4, HC1 andHNO3; strong bases include alkali metal and tetraalkylammoniumhydroxides.

Methanol and ethanol. The self-ionization reaction isROH2

+ + RO~. The pHMe range is about 9.5 and the pHEt about 8.3.HC1O4 continues to be a strong acid and HC1 becomes a 'medium strong'acid (pKAMe = 1.2, p# A E t = 2.1). Acetic acid has the values of pKAMe =9.7 and pKABt = 10.4, i.e. it is much weaker than in water. Strong basesinclude the alkali metal alcoholates, as they introduce RO~ anions, theconjugate base of alcohol. Water is a weak base in ethanol (pKA = 0.3).

Liquid ammonia. Self-ionization occurs according to the equation2NH3*±NH4

+ + NH2~ and the pH range at -60°C is 32 units. Acids thatare weak in water, such as acetic acid, are strong in ammonia and acids thatare very weak in water become medium strong in ammonia. Strong basesinclude potassium amide, introducing NH2~ ions, while hydroxides are weakbases (a KOH solution with a concentration of 10~2 mol dm"3 has a pHNH3

value of about 24.5).If an amphiprotic solvent contains an acid and base that are neither

mutually conjugate nor are conjugated with the solvent, a protolyticreaction occurs between these dissolved components. Four possible situa-tions can arise. If both the acid and base are strong, then neutralizationoccurs between the lyonium ions and the lyate ions. If the acid is weak andthe base strong, the acid reacts with the lyate ions produced by the strongbase. The opposite case is analogous. A weak acid and a weak baseexchange protons:

Strong acid + strong baseWeak acid + strong base

Strong acid + weak base

Weak acid 4- weak base

H2S+ + S - - > ;

HA + S "->i

H2S+ + B ^ 1

HA + B-»]

>HS\ " + HS

^B + + HS

HB+ + A"

(I)

(II)

(HI)(IV)

(1.4.19)

(Except for the last, these reactions are used in titrimetric neutralizationanalysis.) Reactions (II) to (IV) can also proceed in the opposite direction.This will be demonstrated on the well-known example of salt hydrolysis.

Reaction (II) could be the neutralization of acetic acid by potassiumhydroxide, yielding potassium acetate which can be isolated in the crystal-line state. On dissolution in water the K+ cation is only hydrated in solutionbut does not participate in a protolytic reaction. In this way, the weak baseCH3COO~ is quantitatively introduced into solution in the absence of anequilibrium amount of the conjugate weak acid CH3COOH. Thus

54

CH3COO~ reacts with water to form an equilibrium concentration ofCH3COOH molecules, corresponding to the K'A value for acetic acid inwater:

CH3COO- + H2O <± CH3COOH + OH", [CH3COOH][OH-] _ K^ a2

Hc (1.4.20)KH " [CH3COO-] ~ KA(CH3COOH) " 1 - aH

(the degree of hydrolysis of the salt present at the concentration c isorH = [OH-]/c = [CH3COOH]/c, [CH3COO"]/c = 1 - aH.) Because of thishydrolysis reaction, the solution becomes alkaline. For small degrees ofhydrolysis (ar H «l) ,

(L4.21)

The concentration of oxonium ions increases with increasing strength of theacid and the pH decreases.

The case of the salt of a weak base and strong acid is treated in ananalogous way. With decreasing strength of the base, the concentration ofoxonium ions increases and the pH decreases.

In the case of the salt of a weak acid and a weak base HAB, e.g.ammonium acetate, hydrolysis of both the ions occurs:

2HAB + 2H2O ±± A + HA + BH+ + B + H3O+ + OH~

For a high degree of hydrolysis [HA] ~ c and [B] ~ c, where c is the overallconcentration of the salt. The electroneutrality condition gives [A~] +[OH~] = [H3O

+] + [BH+]. Here the concentrations [A"] and [BH+] areexpressed by means of the acid dissociation constants Kf

A(HA) andK'A(BH+) and the oxonium concentration. Finally, we have

[ H 3 ° ]=

pH = i[pK'A(HA) + pK'A(BH+)] (1.4.22)

Thus, the relative strengths of the acid (small pKA(HA)) and of the base(large p#A(BH+)) decide whether the solution becomes acidic or alkaline.

Sometimes, hydrolysis leads to complete transformation of the dissolvedsubstance, for example BC13 + H2O ^B(OH) 3 + 3H3O

+ 4- 3 d " (the solu-tion is acidified by this reaction). Similarly, ammonolysis can occur, forexample BC13 + 6NH3^±B(NH2)3 + 3NH4

+ + 3C1" (the solution is againacidified and excess lyonium ions NH4

+ are formed); fluorolysis can alsotake place, for example UO2F2 + 6HF^±UF6 + 3H3O+ + 2F" (the solutionbecomes alkaline as a result of the formation of lyate ions, F~).

Not only addition of acidic species supplying the ions, H2S+, but also ofacceptors of S~, typically of metal ions Mn+, results in a shift of the

55

protolytic equilibrium towards the H2S+ ions, connected with consecutive

complex formation,1)+ + H2S

+

MS2"-2)+ + H2S+

etc.

Thus, the cation acts as a polybasic acid. Another case of consecutivecomplex formation occurs if the solution contains a complexing agent, X,

(1.4.24)

with consecutive stability constants defined as

[MX,]KMX{ = 77777 7777T (1.4.25)

[MX][X]for i = 1, . . . , r (the charge of the species is not indicated). The more stablethe complex, the larger is the stability constant. Some examples are given inTable 1.8.

Solutions formed by incomplete neutralization of a weak acid by a strongbase or a weak base by a strong acid (cases (II) and (III) in scheme 1.4.19)are called buffers and have interesting properties which are particularlyimportant in many fields of chemistry and biology. Aqueous solutions will

Table 1.8 Consecutive stability constants, expressed as \ogKMXj, ofcomplexes of ammonia (A), ethylene diamine (B), diethylenetriamine(C) and the anion of ethylenediaminetetraacetic acid (D4~) at 20°C and0 . 1 M K N O 3 as indifferent electrolyte. (According to J. Bjerrum and G.

Schwarzenbach)

MAMA2

MA,MA4

MA,MA6

MBMB2

MB,MCMC2

MD2"

Mn2+

—————2.82.10.9——

14.02

Fe2+

—————4.43.32.0——

14.33

Co2+

2.051.570.990.700.12

-0.146.04.93.28.16.0

16.31

Ni2+

2.752.201.691.150.71

-0.017.96.64.710.78.2

18.62

Cu2+

4.133.482.872.11——

10.89.40.1

16.05.3

18.80

Zn2+

2.272.342.402.05——6.05.21.88.95.5

16.50

Cd2+

2.602.051.390.88

-0.32-1.66

5.74.62.18.45.4

16.46

56

be considered by way of illustration. Consider an aqueous solution of aweak acid HA of concentration s = [HA] + [A~], to which potassiumhydroxide is added so that its final concentration in solution is b = [K+]. TheK+ ions do not participate in the protolytic process and OH~ ions presentreact with the acid HA to form water molecules and the conjugate base A~(classically viewed, the salt KA). This reaction is practically quantitative,because OH~ ions are a strong base in aqueous medium. Thus we can set[A~] ~ 6 , i.e. the A~ anions formed by dissociation of the acid HA can beneglected. Further, [HA] = s - [A~]~s - b, so that K'A = [U3O

+]b/(s - b),which yields

pH = pA^(HA) + (1.4.26)

(see Fig. 1.9).The case of a buffer consisting of a weak base and its acidic form (for

example, NH3 and NHJ") is treated in an analogous way. Equations of thetype of (1.4.26) are sometimes called the Henderson-Hasselbalch equations.

Buffering properties are exhibited by mixtures of weak acids (or bases)and their salts with a composition such that the concentration of the weakcomponent and the salt are not too far apart. Thus, for b = \s, pH =PATA(HA). The addition of a quite large amount of strong hydroxide or acidto such a mixture results in a small pH change in the solution (see Fig. 1.9).

The buffering ability of a buffer solution is characterized by its bufferingcapacity /?, which is a differential quotient indicating the change of

0.10

?E 0 . 0 8 -

1* 0 . 0 6 -

0 . 0 4 -

0 . 0 2 -

1 1 >I

; /

IS. I I

1 :.14"Pl(Bi V _8 10 12

pH14

Fig. 1.9 Dependence of pH on the buffer composition according to theHenderson-Hasselbalch equation: (1) acidic buffer (Eq. 1.4.26); (2)

basic buffer. Calculation for K'A = K'B = 1(T4, s = 0.1 mol • dm 3

57

I r

Fig. 1.10 Dependence of the buffering capacity ft on thepH of an acid buffer. Calculation according to Eqs

(1.4.27) for K'A = 10"4, s = 0.01 mol • dm 3

concentration of added strong acid or base in the solution, which isnecessary for a given change of pH (Fig. 1.10). On differentiation of(1.4.26) we obtain

dfrdpH

10 (1.4.27)

Thus, the buffering capacity depends on the composition of the buffer, i.e.on the concentration of the salt a or b. The maximum value found bydifferentiation of Eq. (1.4.27) with respect to a corresponds, for an acidicbuffer, to b = \s.

For practical purposes, a number of buffer mixtures have been proposedthat are useful for various pH ranges. Procedures for their preparation andrequirements on purity and definition of the chemicals used are given inlaboratory manuals and tables.

If the solvent is not protogenic but protophilic (acetone, dioxan,tetrahydrofuran, dimethylformamide, etc.), self-ionization obviously doesnot occur. Consequently, the dissolved acids are dissociated to a greater orlesser degree but dissolved bases do not undergo protolysis. Thus, there canexist only strong acids but no strong bases in these solvents. The pH is notdefined for a solution that does not contain a dissolved acid (i.e. in the puresolvent or in the solution of a base). The pKA value can be defined but not

58

the pKB value, and there is no neutral point. If the solvent is neitherprotogenic nor protophilic, then neither dissolved acids nor dissolved basesare protolysed. Protolytic reactions can occur in such a solvent, but onlybetween dissolved acids and bases (that are not mutually conjugate).

If the dielectric constant of an amphiprotic solvent is small, protolyticreactions are complicated by the formation of ion pairs. Acetic acid is oftengiven as an example (denoted here as AcOH, with a relative dielectricconstant of 6.2). In this solvent, a dissolved strong acid, perchloric acid, iscompletely dissociated but the ions produced partly form ion pairs, so thatthe concentration of solvated protons AcOH2

+ and perchlorate anions issmaller than would correspond to a strong acid (their concentrationscorrespond to an acid with a pK'A of about 4.85). A weak acid in acetic acidmedium, for example HC1, is even less dissociated than would correspondto its dissociation constant in the absence of ion-pair formation. Theequilibrium

HC1 + AcOH«± AcOH2+Cr *± AcOH2

+ + Cl" (1.4.28)

is established. If the constants

[AcOH2+Cr]

A; =

a s s

[HC1]

[AcOH2+Cl~]

[AcOH2+][Cr]

[AcOH2+][Cr]

[HC1] + [AcOH2+Cr]

are introduced, then

(1.4.30)

The protolytic reaction between two molecules must be formulated as theformation of the pair

B + HA^BH + A~ (1.4.31)

that can react as a base with another acid HA,, or as an acid with anotherbase Bn:

B,H+Ar + HAn <P* B,H+A,r + HA,

BjH+Ar + Bn^BuH+Ar + Bj ' '

Protolytic reactions can also occur in fused salts. The solvent participatesin these reactions provided that at least one of its ions has protogenicand/or protophilic character. An example of a solvent in which the cation isaprotic and the anion protophilic is ethylpyridinium bromide (m.p. 114°C).The acid HA is protolysed in this solvent (HA + Br~^HBr + A").Hydrogen bromide acts as a solvated proton and the acidity is expressed as

59

pHBr = — log [HBr] and can be measured with a hydrogen electrode. Inethylammonium chloride (m.p. 108°C) the cation is protogenic and self-ionization occurs, C2H5NH3

++ Cl~ <=±C2H5NH2 +HC1, with the ion prod-uct 10"8'1. Fused alkali-metal hydroxides with amphiprotic OH~ anion arethe most important of these solvents; the autoprotolysis follows theequation 2OH~~«^H2O + O2~. Here, water is the strongest acid, cor-responding to solvated protons, and O2~ is the strongest base. Metal cations(added as hydroxides) behave as acids as they remove O2~ anions, formingcompounds that often precipitate.

So far acid-base reactions of molecules in the ground state of energy havebeen discussed. The properties of these molecules can change considerablyin the electronically excited state (K. Weber, T. Forster, A. Weller, et al).For example, in aqueous solution in the ground state, 2-naphthol haspK'A = 9.5, while the value for the excited state is 2.5. Similarly for phenol,in the ground state, pK'A = 10.0, while the value for the excited state is 3.6.Thus, the strength of acids in these and a number of other cases increases byseveral orders on excitation. Simultaneously, the pH of a solution calculatedfrom the concentrations of species corresponding to the conjugate pairdecreases by several pH units. This effect is termed the laser pH jump'.The effect is studied by irradiating the solution with a short intense pulse(usually provided by a laser—hence the name) and fast fluorescence orabsorption spectroscopy is employed to determine the concentrations of thecorresponding excited acid species and the conjugated base. If the spectro-scopy is sufficiently resolved in time (with resolution of the order ofnanoseconds to picoseconds), the kinetics of the exchange of a protonbetween the dissolved conjugate pair and the solvent can be studied. It hasbeen found that this is a very fast reaction (with a rate constant of the orderof 109s-!)-

1.4.4 Generalization of the concept of acids and bases

The most extensive generalization of the concept of acids and bases wasthat proposed in 1938 by G. N. Lewis, who concluded that the Br0nstedconcept of limiting the group of acids to substances with a dissociablehydrogen prevents the general understanding of the chemistry of acids, inthe same way as the early definition of oxidants as oxygen-containingsubstances impeded the general understanding of oxidation-reductionreactions. In view of this, he related the acidic and basic character ofsubstances to their electron structure by the following definition:

A base is a substance capable of acting as an electron-pair donor.An acid is a substance capable of acting as an electron-pair acceptor.

According to this definition the group of bases practically includes justabout the same substances as the Br0nsted conception, but with a fewadditions such as the inert gases (e.g. argon reacts as a base with borontrifluoride). On the other hand, the category of acids is much wider than

60

under the Br0nsted definition, as it includes species such as H+, HCl, A1C13,SnCl3, SO2, SO3, O, Ag+, Be2+, Cu2+, etc.

The Lewis acids that also fulfil the Br0nsted definition form a specialgroup of protonic acids and can be formally considered to be the products ofneutralization of a proton by a base (for example Cl~).

The reaction between an acid and a base produces a coordinationcompound (complex). In contrast to former theories, this is more anaddition than an exchange reaction. On the other hand, an exchangereaction in the Lewis sense is the result of competition between twoacceptors for a single donor or between two donors for a single acceptor.Thus the typical Br0nsted acid-base reaction HA + B«^A + HB is theresult of competition between two bases :A and :B for the acceptor H+. Thereactions (A1C13 <- A) + :B <± (A1C13 <-B) + :A are analogous. The classicalreactions depend on replacement of one base (Bx) by another (Bn):

HCl + NH3 -* NH4+ + Cl" Salt formation

HCl + H 2 O ^ H3O+ + C r Dissociation (1.4.33)

H2O + NH3 -* NH4+ + OH" Dissociation

H2O + CH3COO~ -* CH3COOH 4- OH" Hydrolysis

The Lewis concept permits inclusion of acids and oxidants in a singlegroup of electrophilic substances and bases and reductants in a single groupof nucleophilic substances. The only difference between acid-base andredox reactions lies in the nature of the bond formed.

However, a wide generalization usually brings a number of complications.The reactions between acids and bases have become so specific that theyevade mutual comparison and arrangement in a series according to theirstrengths. Consequently, Lewis complemented his original definition byfurther, purely phenomenological definitions that are not in any way relatedto the theory of the structure of molecules and are based only on thebehaviour of substances during chemical reactions:

Bases are substances that (similar to OH~ ions) neutralize hydrogen ionsor some other acid.Acids are substances that (similar to H+ ions) neutralize hydroxide ionsor other bases.

Acids and bases have the following properties:

(a) The mutual neutralization reaction is fast and has low activation energy.(b) Species of the same type can exchange with one another (e.g. one base

for another one).(c) They can be titrated with the help of acid-base indicators.(d) They catalyse chemical reactions.

These complementary definitions, however, do not solve the problems, as

61

some of substances exhibit only one of the features required, whilst lackingothers, and thus their placing in the acid or base group is not certain.

The concept of 'hard' and 'soft' acids and bases ('HSAB') should also bementioned here. This is not a new theory of acids and bases but represents auseful classification of Lewis acids and bases from the point of view of theirreactivity, as introduced by R. G. Pearson.

Hard acids (HA) have low polarizability and small dimensions, higheroxidation numbers and the 'hardness' increases with increasing oxidationnumber; they bind bases primarily through ionic bonds. Typical examplesare H+ , Na+, Hg2+, Ca2+, Sn2+, VO2+, VO2

2+, (CH3)2Sn2+, A1(CH3)3, I7+,

I5+, Cl7+, CO2, R3C+, SO3.Soft acids (SA) are strongly polarizable, have low or zero oxidation

numbers, the 'softness' decreases with increasing oxidation number and ifgroups with a large negative charge are also bonded to the central atom,then the 'softness' increases; they bind bases primarily through covalentbonds. Typical examples are Cu+, Ag+, Hg+, Cs+, Pd2+, Cd2+, Hg2+, Tl3+,T1(CH3)3, BH3, I

+, Br+, HO2+, I2, Br2, O, Cl, Br, I, metal atoms, metals in

the solid phase.Hard bases (HB) have low polarizability and strong Br0nsted basicity; in

the V, VI and VII groups of the periodic table, the first atom is always thehardest and the hardness decreases with increasing atomic number in thegroup. Typical examples are H2O, OH~, F", O2", CH3COO~, SO4

2~, Cl",CO3

2", NO3", ROH, RCT, R2O, NH3, RNH2, N2H4.Soft bases (SB) are strongly polarizable and generally have low Br0nsted

basicity. Typical examples are R2S, RSH, RS", I~, R3P, CN", CO, QH4,C6H6, H , R .

Obviously, a number of substances cannot be classified exactly. Border-line acids include Fe2+, Sb3+, Ir3+, B(CH3)3, SO2, NO+, R3C+ and C6H5

+

and borderline bases include C6H5NH2, N3~, Br~, NO2", and SO2~.Simple laws govern the reactions of these substances: hard acids react

preferentially with hard bases and soft acids with soft bases. Mixing of'unsymmetrical' complexes results in the reaction

HA—SB + SA—HB-^ HA—HB + SA—SB. (1.4.34)

For example, the ether RO—C^ (of the HB—SA type) is split by hydrogeniodide (of the HA—SB type) to yield ROH (the HB—HA type) and I—G^(the SB—SA type).

It is obvious that the Lewis theory and the HSAB concept are veryimportant for the description of the mechanisms of chemical reactions;however, for electrochemistry the Br0nsted theory is quite adequate.

1.4.5 Correlation of the properties of electrolytes in various solvents

It has already been mentioned several times that the strength of acids andbases is expressed by their dissociation constants KA and KB. This is

62

actually true if acids or bases are compared in a single solvent. For example,in aqueous solutions values of KA of 1.38 x 1(T3, 1.74 x 10~5 and 4.27 x10~7 have been found for chloroacetic acid, acetic acid and carbonic acid,respectively; consequently, in (dilute) aqueous solutions chloroacetic acid istwo orders of magnitude stronger than acetic acid which is again two ordersstronger than carbonic acid. These values have no absolute significance.They change on transfer to another solvent and even their order canchange. If, for example, the protolytic reaction CH3COOH + H2O <=*CH3COO~ + H3O+ occurred in acetic acid medium, with water as adissolved base, then the standard state to which the activities andequilibrium constants are related would be changed. Comparison of theacidity or basicity of solvents is just as difficult. For example, alcohols reactaccording to the equation ROH +H3O+^±ROH2

++ H2O. In dilute solu-tions (water concentration of less than 1 mol • dm"3), the constant K' forthis reaction in methanol is equal to 8.6 x 10~3 and in ethanol, 3.9 x 10~3. Ifthese values were also valid in the vicinity of composition [ROH]/[H2O] =1, then the ratio [ROH2

+]/[H3O+] = /T[ROH]/[H2O] describing the protondistribution between the alcohol and water would be small. Consequently,water would have a greater affinity for protons than alcohol, i.e. would bemore basic. The opposite is true.

The effect of the medium (solvent) on the dissolved substance can best beexpressed thermodynamically. Consider a solution of a given substance(subscript i) in solvent s and in another solvent r taken as a reference.Water (w) is usually used as a reference solvent. The two solutions arebrought to equilibrium (saturated solutions are in equilibrium when each isin equilibrium with the same solid phase—the crystals of the dissolvedsubstance; solutions in completely immiscible solvents are simply broughtinto contact and distribution equilibrium is established). The thermo-dynamic equilibrium condition is expressed in terms of equality of thechemical potentials of the dissolved substance in both solutions, jU,(w) =fii(s)y whence

RT In = tf(s) - p?(w) = AG&rg (1 -4.35)

where jU°(s) and jU°(w) are the standard chemical potentials, identical withthe standard Gibbs energies of solvation. The quantity AG?r '^

s is thechange in the partial molar Gibbs energy of transfer of the dissolvedsubstance from the standard state in the reference solvent, w, to thestandard state in the solvent considered, s, and is termed the standard Gibbstransfer energy. A negative value of the standard Gibbs energy of transferfor a proton, AG&jJ?8 indicates that a proton is more stable in solvent s anda positive value indicates that it is more stable in water.

Some authors prefer the coefficient y^Js to the transfer Gibbs energy; this

63

coefficient is defined by the relationship

TF) (iA36)

This coefficient has various names (medium effect, solvation activitycoefficient, etc.); the name recommended by the responsible IUPACcommission is the transfer activity coefficient. In this book the effect ofsolvation in various solvents will be expressed exclusively in terms ofstandard Gibbs transfer energies.

In electrolytes, the ionic and the overall Gibbs transfer energies must bedistinguished. These quantities are defined in the usual manner. Forexample, for the most usual type of electrolyte AB—»A + B,

AG?r',lT = A G £ r 8 + AG?r;Vs (1.4.37)

The Gibbs energy of an ion changes on transfer from one solvent toanother primarily because the electrostatic interaction between ions and themedium changes as a result of the varying dielecric constant of the solvent.This can be expressed roughly by the Born equation (see Eq. 1.2.7),

i _ i\( 1 A 3 8 )

Mostly, Ds < Dw so that the Gibbs energy increases on transfer from waterto some other solvent.

The determination of the standard Gibbs energies of transfer and theirimportance for potential differences at the boundary between two immis-cible electrolyte solutions are described in Sections 3.2.7 and 3.2.8.

1.4.6 The acidity scale

An exact description of the acidity of solutions and correlation of theacidity in various solvents is one of the most important problems in thetheory of electrolyte solutions. In 1909, S. P. L. S0rensen suggested thelogarithmic definition of acidity for aqueous solutions considering, at thattime, of course, hydrogen instead of oxonium ions (cf. Eq. (1.4.11))

pH = -log a(H3O+) = -log [H3O+] - log y(H3O+) (1.4.39)

This notional definition of the pH scale can, however, not be used forpractical measurements, as it contains the activity coefficients of theindividual ions, y(H3O+).

Under these conditions, an operational definition must be used, i.e. adefinition based on the method by which the quantity pH is measured.Obviously, the operational definition must be formulated as closely aspossible to the 'absolute' equation (1.4.39). In this way, a practical scale ofpH is obtained that is very similar to the absolute scale but is not identical

20Weight %H 2 S0 4

Fig. 1.11 Dissociation ranges of colour indicators for determination of the acidity function Hoin H2SO4-H2O mixtures: (1) p-nitroaniline, (2) o-nitroaniline, (3) p-chloro-o-nitroaniline, (4)/7-nitrodiphenylamine, (5) 2,4-dichloro-6-nitroaniline, (6) p-nitroazobenzene, (7) 2,6-dinitro-4-methylaniline, (8) benzalacetophenone, (9) 6-bromo-2,4-dinitroaniline, (10) anthraquinone,

(11) 2,4,6-trinitroaniline. (According to L. P. Hammett and A. J. Deyrup)

65

with it. This definition has changed somewhat over the years and thedefinitions have now been unified (Pure Applied Chemistry, 57, 531, 1985).As these definitions are based on pH measurement with the hydrogenelectrode they will be discussed in Section 3.3.2. For the same reason thepH scales in non-aqueous solutions will be dealt with in Section 3.2.6.

For solutions of extreme acidity (e.g. the H2O-H2SO4 mixture) theacidity function Ho was introduced by L. P. Hammett as

Ho = pATA(BH+) + log - ^ - = -log aH+ - log - ^ - (1.4.40)C B H + YBH+

Here, KA(BH+) denotes the acid dissociation constant of a basic colouredindicator B (for the theory, see Section 1.4.7).

If a given indicator is half dissociated, the value of Ho is equal to thedecadic logarithm of its acidity constant times — 1. For practical measure-ments an indicator may be used only in the range of its colour transition,i.e. at most in the range of the ratio cB/cBH+ from 10 to 10"1. For a largerinterval of Ho values several indicators have to be used.

In view of the term containing activity coefficients, the acidity functiondepends on the ionic type of the indicator. The definition of Ho is combinedwith the assumption that the ratio YB/YBH+ is constant for all indicators ofthe same charge type (in the present case the base is electroneutral; hencethe index 0 in //0). Thus, the acidity function does not depend on eachindividual indicator but on the series of indicators.

The acidity function is determined by successive use of a range ofindicators. Hammett and Deyrup started with /?-nitroaniline, the pKAx ofwhich in a dilute aqueous solution is 1.11 (the solvolysis constant has beenidentified with the acidity constant). Since in a dilute aqueous solution,7B

= YBH+ ~ 1 > the acidity function for the aqueous media goes over to the

pH scale. By means of p-nitroaniline, the acidity constant of anothersomewhat more acidic indicator is obtained under conditions such that bothforms of each indicator are present at measurable concentrations. Then Ho

as well as the pKAl of the other indicator is determined by using Eq.(1.4.40). By means of this indicator, values of Ho not accessible with/?-nitroaniline may be reached. The Ho scale is further extended by using athird indicator and its pKA3 is determined in the same way (see Fig. 1.11).The concentration ratios are determined photometrically in the visual orultraviolet region. Figure 1.12 shows the dependence of Ho on the com-position of the H2SO4-H2O mixture and was obtained as indicated above.

1.4.7 Acid-base indicators

The interest in colour indicators has recently increased as they are usedfor the direct determination of pH (acid-base indicators) and free calciumions (fluorescent derivatives based on the calcium chelator EGTA asmetallochromic indicators) in biological systems at cellular level.

66

i l l

20 60Weight °/oH2S04

100

Fig. 1.12 Acidity function H() for the mixtureH2SO4-H2O as a function of the composition.

(According to L. P. Hammett and M. A. Paul)

While the role of acid-base indicators in acidimetric titrations has beensurpassed by more advanced, automatic, potentiometric methods variousmetallochromic indicators are still in use in complexometric analysis. Herewe shall deal only with acid-base indicators.

According to the early view of Ostwald, acid-base indicators are weakacids or bases, the undissociated form of which differs in colour from theionic form. For example, the molecule of an indicator HI dissociates inwater according to the equation

HI + H2O = H3O+ + I" (1.4.41)

The equilibrium constant of this reaction in a dilute solution is

fH3O+][I-]

whence

[HI]

pH = pKi + log[HI]

(1.4.42)

(1.4.43)

showing the dependence of the ratio [I ]/[HI] on pH.

67

In a later conception due to Hantzsch, an indicator possesses twotautomeric forms differing in colour, and at least one form functions as anacid or base. If both tautomeric forms are capable of dissociation their ionsalso differ in colour. However, the ions and molecules of the same formshow identical light absorption because such a small variation of thestructure of a molecule as the dissociation of a proton cannot cause a largechange in molecular property such as light absorption. For example,phenolphthalein does not absorb visible light in the acid and neutral region,but at pH > 8.2 one of its hydroxyl groups becomes ionized and, at the sametime, a tautomeric change takes place:

)H

(1.4.44)

Colourless form Red form

The quinoid group

is a chromophore which absorbs light radiation in the whole visible region ofthe spectrum except for the long wavelength range (red light). Onabsorption by the quinoid system the light energy is dissipated in the formof the energy of thermal movement of solvent molecules. However, in thecase of fluorescence indicators this energy is converted to light of differentwavelength which is recorded with a fluorescence spectrometer.

In the case of only one form of the indicator undergoing the tautomericchange to a coloured form (this is the case of phenolphthalein anion) theequilibrium between the original form of the indicator anion and itstautomeric form is characterized by the tautomeric constant

KT = | ^ | (1.4.45)

where I" is the coloured (in the case of phenolphthalein the quinoid) formof the anion of the indicator. For K » 1

[I-]=s (1.4.46)where s is the overall concentration of the indicator. Thus, when substitut-ing for [HI] =5 — [I~] into Eq. (1.4.41) an analogy of the Henderson-Hasselbalch equation (1.4.26) is obtained, which is the basis of thedetermination of pH by means of acid-base indicators.

68

The determination of intercellular pH is based on measurement offluorescence emitted by the fluorescence indicator, fluorescein. The startingcompound used for this purpose is fluorescein diacetate:

O—C—CH,

Fig. 1.13 pH maps of Saccharomyces cerevisiaecell transferred from water to a 0.2 M buffer ofpH 3: (a) after 2min; (b) after 5 min; (c) after20min. (Reproduced by permission of J. Slavik

and A. Kotyk)

69

This substance penetrates into the cell where it is hydrolysed to fluoresceinthrough the action of the enzyme's esterases. The quantity of the fluorescentquinoid form then depends on the local value of pH in the same way as inthe case of phenolphthalein. The intensity of the fluorescein radiation ismeasured with a fluorescence microscope and then processed to a digitalimage which is the basis of a map of pH distribution in the cell (Fig. 1.13).

References

Albert, A., and E. P. Sergeant, The Determination of Ionization Constants. ALaboratory Manual, 3rd ed., Chapman & Hall, London, 1983.

Bates, R. G., Determination of pH. Theory, Practice, John Wiley & Sons, NewYork, 1973.

Burger, K., see page 27.King, E. J., Acid-base behaviour, in Physical Chemistry of Organic Solvent Systems

(Eds A. K. Covington and T. Dickinson), Chap. 3, Plenum Press, New York,1973.

Lengyel, S., and B. E. Con way, Proton solvation and transfer in chemical andelectrochemical processes, CTE, 5, Chap. 4.

Rochester, C. R., Acidity Functions, John Wiley & Sons, New York, 1970.Slavik, J., in press.Tremillon, B., see page 27.

1.5 Special Cases of Electrolytic Systems

1.5.1 Sparingly soluble electrolytes

A great many electrolytes have only limited solubility, which can be verylow. If a solid electrolyte is added to a pure solvent in an amount greaterthan corresponds to its solubility, a heterogeneous system is formed inwhich equilibrium is established between the electrolyte ions in solution andin the solid phase. At constant temperature, this equilibrium can bedescribed by the thermodynamic condition for equality of the chemicalpotentials of ions in the liquid and solid phases (under these conditions,cations and anions enter and leave the solid phase simultaneously, fulfillingthe electroneutrality condition). In the liquid phase, the chemical potentialof the ion is a function of its activity, while it is constant in the solid phase.If the formula unit of the electrolyte considered consists of v+ cations andv_ anions, then

v+iu+(s) + v_]U_(s) = v+fi°+(\) + v_^( l ) 4- v+RT In a+ + v_RT In a.

(1.5.1)

from which it follows that

, ? « : - - P - exp {V-'"° <*> - *°<'>' + " - ' " - < S > ' *° <•>') (1.5.2)I RT J

The product P, termed the (real or thermodynamic) solubility product, is aconstant dependent only on the temperature. The analogous (apparent)

70

solubility product P', defined in terms of concentrations or molalities,depends on the ionic strength. Obviously

P = P'yl (1.5.3)

At ionic strengths where the activity coefficient decreases with increasingionic strength, the product P' increases because the product P is constant.

The solubility, csat, is the concentration of the saturated solution and thusdepends on the product P'. As c+ = v+csat and c_ = v_csat, then

< L 5 - 4 >

If the concentration of one of the ions is increased (for example c_) byaddition of its soluble salt (for example KC1 is added to a saturated AgClsolution), then the solubility of the electrolyte, AgCl, as a whole is given bythe concentration of the ion present in the lower concentration (csat =c+/v+). It can readily be seen that logcsa t=-(v_/v+) logc_ +(l/v+) log P — log v+. The logarithm of the solubility thus decreases linearlywith the logarithm of the increasing concentration of the added ion. Theseparations by precipitation in analytical chemistry is based on this principle.

Examples of solubility products are listed in Table 1.9. Similarly to thedissociation constants, the solubility products are also dimensionless quan-tities. However, because of the choice of the standard state, their valuesnumerically correspond to units of moles per cubic decimetre.

The solubility product is measured by determining the ion concentrationsby a suitable analytical method and the results are extrapolated to zero ionicstrength, where P' = P.

1.5.2 Ampholytes

Ampholytes are substances which, according to the acidity of themedium, act either as acids or as bases. Such a case has already been dis-

Table 1.9 Solubility products P of some sparingly soluble salts. (According to B. E.Conway)

Salt

CdSCa(COO)2

CoSCuSCu2SCu2I2

Mg(OH)2

Hg2Cl2

Hg2Br2

'CO2525202525182519.219.2

P

1.14 x1.78 x

3 x3 x

8.5 x5.0 x4.6 x

5.42 x3.9 x

io- 2 8

lO'9

io- 2 6

1Q-38

io-4 5

10'1 2

lO' 2 4

10'1 9

10'2 3

Salt

Hg2I2

Hg2CrO4

AgClAgBrAglAg2SAgCNST1C1Til

'CO19.22525252525252525

P

1.05 x2.0 x1.8 x6.5 x1.0 x

3 x1.44 x2.25 x6.47 x

101010101010101010

- 1 9

- 9

- 1 0

- 1 3

- 1 6

- 5 2

- 1 2

- 4

- 8

71

cussed as the anions of a two-protonic acid possess amphiprotic properties.For example, the hydrogen sulphite anion reacts in two ways: in alkalinemedium as an acid, HSO3~ + H2O^SO3

2~ + H3O+, while in acidic me-dium as a base, HSO3~ + H2O«=*H2SO3 + OH". As an ampholyte, how-ever, such a substance is usually considered whose neutral molecule HP hasboth the acidic and basic functions. In an aqueous solution the followingequilibria are established:

HP + H2O*±H3CT + P-H2P+ + H2O^±H3O+ + HP l ' ' j

with acid dissociation constants

Kx=—- and K2 = — (1.5.6)aHP « H 2 P +

This simple treatment is applicable to cases where HP is a real neutralmolecule. Although typical examples of ampholytes are amino acids, inwhich case the equilibrium (1.5.5) would be written

NH2RCOOH + H2O «± H3O+ + NH2RCOO~

NH3+RCOOH + H2O *±NH2RCOOH + H3O+ ^"5"7^

However, in aqueous solutions the amino acids will have undergone internalionization:

NH2RCOOH<=>NH3+RCOCT (1.5.8)

This equilibrium lies almost completely to the right-hand side, so that theNH2RCOOH molecules are practically absent from the solution. A particleof the type NH3

+RCOO~ is called an amphion (zwitterion). The internalionization is proven by several empirical facts:

(a) The amino acids have a large dipole moment.(b) The acidic dissociation reactions described by (1.5.6) and (1.5.7) have

the constants Kl ~ 10"10 and K2~ 10~2, whereas based upon values foracetic acid and ammonia, the constants should be of the order of 10~5

and 10~9 for glycine; this difference cannot be attributed to thereplacement of hydrogen in the acetic acid molecule by the amine groupor, to replacement of hydrogen in the NH3 molecule by the residue ofacetic acid, CH2COOH.

(c) By blocking the basic functional group (e.g. in the reaction of glycinewith formaldehyde) the acidity increases strongly and approaches thatof the fatty acids.

In view of this, the protolytic equilibria may be formulated as

NH/RCOCT + H2O^NH2RCOCT + H3O+

NH3+RCOOH + H2O *± NH3

+RCOO" + H3O+ ^" 5 ' ^

72

with acid dissociation constants

flNH3+RCOOH

From (1.5.10) we may conclude that the constant Kx does not charac-terize the acidity of the carboxyl group but of the NH3

+ group, and viceversa for K2. The deviations from the accepted values of the aciddissociation constants of acetic acid and ammonium ion may be attributed tothe influence of the constitution of the ionic forms of the amino acid. Theincrease in acidity of the carboxyl group is caused by the neighbouringpositive charge of —NH3

+, while the negative charge of —COO" is thecause of the increase in basicity of the amine group. The values of thedissociation constants K[ and K2 are listed in Table 1.10.

An important quantity characterizing the amino acids, peptides andproteins is the isoelectric point, which is the value of pH at which theampholytes do not migrate in an electrical field. This occurs when

WNH3+RCOOH = mNH2RCOO- (1.5.11)

From (1.5.5) we obtain for a dilute solution

„ v 2 mNH2RCOQ-7NH2RCOO- ^H3O+^NH2RCQQ- / 1 c 1->\

KxK2 = aU3O+- « — - (1.5.12)mNH3

+RCOOH7NH3+RCOOH mNH3

+RCOOH

Using the condition (1.5.11) for the isoelectric point we have (subscript /

Table 1.10 Dissociation constants of amino acids at 25°C.(According to B. E. Con way)

Amino acid

GlycineAlaninea-Amino-n -butyric acidValineoc-Amino-n -valeric acidLeucineIsoleucineNorleucineSerineProlinePhenylalanineTryptophaneMethionineIsoserineHydroxyvalineTaurine"jS-Alanine

2.342.342.552.322.362.362.362.392.211.991.832.382.282.782.611.53.60

P*2

9.609.699.609.629.729.609.689.769.15

10.609.139.399.219.279.718.74

10.19

For —SO3H.

73

denoting the isoelectric point)

(1.5.13)

At the isoelectric point, the acid dissociation equals the base dissociation.The degrees of dissociation ocx and oc2 are defined in such a way that

^ N H 3+ R C O O - = S(l - OCX - QC2)

(1.5.14)

)-=sa2

where s is the overall concentration of the ampholyte. From the definitionsof Kx and K2 and from (1.5.14) we obtain

At the isoelectric point or, = #2 = (ar)7, so that on eliminating a from Eqs(1.5.15) we obtain Eqs (1.5.13).

Furthermore, at the isoelectric point the total dissociation, ax + a2, is atits minimum. This sum may be obtained from Eqs (1.5.15) so that ondifferentiating with respect to flH3o

+ we have, for activity coefficients exceptthat of H3O+ equal to unity,

ar2) = 2flH3

H 3 O+

whence, after some rearrangement, Eq. (1.5.13) is again obtained. Thecourse of dissociation is shown schematically in Fig. 1.14.

The degree of dissociation at the isoelectric point can be found bycalculating aH3o

+ from Eq. (1.5.15), putting ocx = oc2 = (a)f«1 and sub-stituting into (1.5.13). In a dilute solution

| (1-5.17)

Assuming the sensitivity of the determination of a at the isoelectric point as1 per cent, the isoelectric point will appear as a real 'point', if (a)I > 10~2,that is KJK2> 10"4, whereas for (or)7<10"2, that is KJK2< IO~4, adefinite interval of pH obeys the isoelectric condition (see Fig. 1.14).

1.5.3 Polyelectrolytes

Polyelectrolytes are macromolecular substances whose molecules have alarge number of groups that are ionized in solution. They are termedmacroions or poly ions and are studied most often in aqueous solutions.

74

1.U

a

0.6

0.2

X • !\ PH

1

SO i

/

/

/

. PKi

-

A _

4 6 8 10 0pH

Fig. 1.14 Dissociation of an ampholyte. (A) pK[ = 8, pA 2 = 6; (B) pK[ = 9,pK'2 = 2. Calculation according to Eq. (1.5.15)

Small counterions (gegenions) ensure electroneutrality. For example, thesodium salt of polyacrylic acid is ionized in solution to form polyanions andthe corresponding number of sodium cations:

'— CH—CH—\

0=0 I +

The fact that one of the ions has large dimensions affects the electrochemi-cal properties of the system compared with the corresponding low-molecularsystem. On the other hand, the presence of an electric charge affects thebehaviour in solution of the macromolecular system compared to thecorresponding system without electric charge.

Two extreme types of polyanions can be distinguished. In the first, themain chain is flexible and can assume a large number of differentconformations so that the overall shape of the ion depends on the solvent.In some solvents (called 'bad'), they are rolled up to form a relatively rigid'random coil', while in others ('good') the coil is more or less expanded.This type includes most synthetic polyelectrolytes such as the followingacids:

Polyacrylic Polymethacrylic

CH3

(_CH 2 -CH-) W (-CH2—C—)„

COOH COOH

75

Polyvinylsulphonic Polystyrenesulphonic

(_CH2-CH-)n (-CH2-CH-), ,

SO3H C6H4

SO3H(the first two are weak in aqueous solutions while the other two are strong).In copolymers of maleic acid with styrene, two acid groups are alwaysseparated by one or more styrene units, so that a dibasic poly acid is formed:

(—CH2—CH— C H — C H )„

C6H5 COOH COOH

Analogously, a tribasic poly acid can be represented by the copolymer ofmaleic acid with acrylic acid:

(—CH2—CH CH CH—)„

COOH COOH COOH

Polymetaphosphoric acid is an example of an inorganic poly acid:

while polyvinylamine is an example of a weak polybase:

( - C H 2 - C H - ) n

NH2

A second type of poly ion assumes stable, specific conformation. Thisgroup includes many natural poly electrolytes. For example, the side chainsof proteins have a large number of acidic and basic ionizable groups. Theresultant sign and amount of charge of the polyion depends on the pH (thespecies being a polyampholyte). The tertiary structure of the globularprotein is so stable that it is not denatured even at a high value of theoverall charge. Another example is the native form of deoxyribonucleicacid, whose double helix is quite stable in solution; the polyion acts insolution as a rigid rod. The difference in behaviour between polyion with afixed shape and that of low-molecular-weight ions in solution depends onlyon their size and their overall charge.

The behaviour in solution of the first type of polyions compared with thatof uncharged molecules is more involved in that the coil expansion isaffected not only by the solvent but also by the electric field formed by thepolyion itself, by the counterions and by the ions of other low-molecular-weight electrolytes, if present in the solution. Infinite charge dilution cannotbe achieved by diluting the poly electrolyte solution, as a local high-intensity

76

Fig. 1.15 A schematic representation of a polyacid solution including polyanions©,anions of an added low molecular electrolyte O and positive counterions #

electric field remains inside the macroion, attracting a corresponding numberof counterions (see the scheme in Fig. 1.15). Thus, the concept of the ionicstrength of the solution lacks sense as an infinitely dilute solution of apolyelectrolyte does not behave ideally at any conditions and cannot beconsidered to be in the standard state. On the other hand, mutual repulsionof charges on the chain can lead to considerable chain expansion, whichcould not be achieved by transfer of the uncharged macromolecules from a'bad' to a 'good' solvent.

The shape of any macromolecule with a flexible chain, both unchargedand charged, i.e. the degree of expanding or rolling-up the random coil, ischaracterized by the parameter h, corresponding to the distance betweenthe ends of the original chain from which the coil is formed. The probabilitythat the ends will lie at a distance h is characterized by the probabilitydensity w(h) derived in textbooks on macromolecular chemistry by statisti-cal methods on the basis of various models with various simplifyingassumptions. In the resultant equations the number of basic segments in themacromolecule, Z, and their length, b> is taken into account and the actualsystem is described more or less accurately (and the equations are more orless complex) depending on the accuracy of the model employed. Forpolyelectrolytes, w(h) consists of the a priori probability wo(h),corresponding to the uncharged macromolecule, and the Boltzmann factor,

77

containing the excess electrostatic Gibbs energy AGf,(/*):

w(h) dh = wo(h) exp ( ~ A ^ l ( / l ) ) d* (1.5.18)

The distribution of distance h is characterized either by the most probablevalue hmax, defined by the condition dw(/i)/d/* = 0, or by the mean squareof distance h, defined in the usual manner, h2 = $h2w(h) dh/$w(h) dh. Theratio (hmax)

2/h2 gives the width of the distribution. The closer this value is tounity, the narrower the distribution. This ratio is equal to \ for the simplestmodel in statistics, the 'random walk'.

The problem is then reduced to determining the energy AGfj, which againdepends on the model. The simplest concept (W. Kuhn, A. Katchalsky, etal.) is to place half of the total charge Q at one end of the polyion and theother half at the other end, from which it follows that AGf,(/i) = Q2/4eh,where e is taken at a first approximation to be the macroscopic permittivityof water. Another greatly simplified model, corresponding to a certaindegree to globular proteins, considers the polyion as a spherically symmetri-cal charge cloud. The electrostatic contribution to the excess Gibbs energyaffects the thermodynamic, rheological, transport, and other properties ofpolyelectrolytes. Here, only dissociation equilibria will be consideredbriefly.

Consider a molecule of a carboxylic polyacid with degree of polymeriza-tion N, bearing N ionizable groups, n of which are actually ionized. K'n isthe dissociation constant for the ionization of an acidic group bound to ann-times ionized molecule; K^ is the dissociation constant of the group thatsplits off the first proton and therefore has not yet been affected by theelectric field of the other already ionized groups. The expression — kT\n KQgives the Gibbs energy change due to ionization of a neutral molecule. Thevalue of —kT\nK'n differs from this expression by the electric worknecessary for removal of a further proton from the field of an n-fold ionizedmolecule. If the electric charge of an rt-times ionized molecule is AG^, thiswork is equal to dG^/dn. Then we have

-kT In K'n = -kT In K'Q + ~ (1.5.19)dn

Except at the first stage in neutralization titration, i.e. for small values of n,all the molecules in solution are found experimentally to be ionized to thesame degree a = n/N. Then

is* rx_f o + i ri-i o + i (*\ < on\

78

Thus, the dissociation equilibrium is affected by the ionic strength,temperature and dielectric constant of the solvent as well as by theparameter h (involved in AGfj). On the other hand, the term dG^/dn doesnot depend on the degree of polymerization (except for very small values ofn). The degree of polymerization does not affect, for example, the course ofthe potentiometric titration of a poly acid.

For a dibasic poly acid, where, of the total of TV units, nx are ionized to thefirst stage and n2 to the second stage, then simultaneously

\niox

r &2(n2+l) l n l O x f l ^ T r u In 10 x kT 3{nx + n2)

Here, /? is the average distance between two neighbouring doubly ionizedmonomeric units and e is the effective dielectric constant.

For a polyampholyte we have

pH = pKlitA - logocA In 10 x kT dnA

\-aB 1 SGaB In 10 x kT

(1.5.23)

where K'QA and K^B are the acid dissociation constants of the monomericunit of the ampholyte, aA = nA/NA and aB = nB/NB, where nA and nB arenumbers of ionized acid and basic groups and NA and AfB are total numbersof ionizable acid and basic groups, respectively. These equations are,however, invalid in the vicinity of the isoelectric point where large attractiveforces act inside the molecules. A result of this is a decrease of the solubilityand coagulation of the polyampholyte.

References

Fasman, G. D. (Ed.), Prediction of Protein Structure and the Principles of ProteinConformation, Plenum Press, New York 1989

Jurnak, F., and A. McPherson (Eds), Biological Macromolecules and Asemblies,Vol. 2, Nucleic Acids and Interactive Proteins, John Wiley & Sons, New York,1985.

Manning, G., Quart Rev. Biophys., 11, 179 (1978).Morawetz, H., Macromolecules in Solution, John Wiley & Sons, New York, 1965.Oosawa, F., Poly electrolytes, M. Dekker, New York, 1971.

Chapter 2

Transport Processes in ElectrolyteSystems

Since water as solvent plays the role of a medium where electrolytic displacementstake place we shall be able to state for sure, together with Wiedemann, Beetz andQuincke, that the electrical resistance of a solution consists of resistances tomovement enforced upon the components of the solution by water particles, by thecomponents themselves and perhaps by the undecomposed molecules of theelectrolyte. To separate these various hindrances will be no easy task, particularlybecause, as stressed by Quincke, they are not necessarily constant but can, forexample, depend on the condition of the solution. Even from this standpoint theprocess of conduction will be, in general, still very complicated.

The discussion becomes different when restricted to dilute solutions. In this caseone must arrive at the conclusion which essentially helps to simplify the phenomena:In a dilute solution the conductivity depends (besides on the number of dissolvedmolecules) only on the transported components being independent of their mutualassociations. Thus, the more the number of water molecules prevails over those ofthe electrolyte the more pronounced is the influence exerted by the water moleculeson the ions and the less their mutual friction.

F. Kohlrausch, 1879

2.1 Irreversible Processes

In a homogeneous medium of an electrolyte solution, an ionic liquid or asolid electrolyte under conditions of constant pressure and temperature,mechanical, electrostatic and short-range forces act on the individualparticles in solution, but these forces average out in time. The effect ofthese forces is reflected in the activity values of the individual componentsof the system.

Let us consider another situation where a force (or forces) is notcompensated on a time average. Then the particles upon which the force isexerted become transported in the medium. This 'translocation' phenome-non changes with time. Particle transport, of course, also occurs underequilibrium conditions in homogeneous media. Self-diffusion is a processthat can be observed and its velocity can be measured, provided that agradient of isotopically labelled species is formed in the system at constantcomposition.

79

80

The forces leading to transport of matter in the system are connected withthe transfer of impulse during collisions between particles, with the effect ofexternal mechanical forces and gravitation on the whole system or partthereof and with the effect of the electrostatic field on the impulse of theparticles. The manifestation of forces defined in this manner in the course oftransport phenomena can be treated quantitatively by statistical mechanics.However, a different approach is usually employed. These uncompensatedmechanical and electrical forces are expressed in terms of 'phenomenologi-cal driving forces'. The response of the system to these forces is thendescribed in terms of the fluxes of the corresponding thermodynamicquantities. The flux (or flux intensity by some authors) is defined as theamount of substance, energy or charge passing through a unit area in a unittime. If a single driving force X acts on one species of particles thecorresponding flux is

= LX (2.1.1)

where L is the phenomenological coefficient, a scalar quantity correspond-ing to a generalized conductance or to the reciprocal of a generalizedresistance. Onsager broadened this simple formulation to include the casewhere the fluxes of n thermodynamic quantities are the result of n differentdriving forces. Assuming that the system is not far from equilibrium, thefluxes J, are linear functions of the driving forces (linear irreversiblethermodynamics):

(2.1.2)k

According to the Onsager assumption, the square matrix of the ph<noiopical coefficients

gnological coefficients

enome-

L22 L2n

(2.1.3)

is symmetrical. Thus, the reciprocity relationship

Lik = Lki (2.1.4)

is valid.Four basic transport phenomena can be distinguished:

1. Diffusion—transport of matter as a result of differing values of thechemical potential of a given component at various sites within thesystem, or in the system and its surroundings. Obviously, the particles

81

present in a given volume at a site of lower concentration have a lowerprobability to come to the same volume at a site of higher concentrationthan in the opposite case, where the particles at a site of higherconcentration come to the site of lower concentration.

2. Heat conduction—the transport of energy resulting from temperaturegradients in the system or difference in temperature, between the systemand its surroundings. Heat conduction arises from the fact that, while thenumber of particles per volume unit is identical at various sites in thesystem, they have different impulses.

3. Convection—the transport of mass or energy as a result of streaming inthe system produced by the action of external forces. These includemechanical forces {forced convection) or gravitation, ifthere are density gradients in the system {natural or free convection).

4. Conduction of electric current—transport of charge and mass as the resultof the motion of elecrically charged particles in an electric field.

Chemical reactions in the system are irreversible processes, affectingtransport processes, as they result in the formation and disappearance ofcomponents of the system and in the release or consumption of thermalenergy.

The rate of a chemical reaction (the 'chemical flux'), /ch, in contrast to theabove processes, is a scalar quantity and, according to the Curie principle,cannot be coupled with vector fluxes corresponding to transport phenom-ena, provided that the chemical reaction occurs in an isotropic medium.Otherwise (see Chapter 6, page 450), chemical flux can be treated in thesame way as the other fluxes.

References

De Groot, S. R., and P. Mazur, Non-equilibrium Thermodynamics, Interscience,New York, 1962.

Denbigh, K. G., Thermodynamics of the Steady State, Methuen, London, 1951.Glansdorff, P., and I. Prigogine, Structure, Stability and Fluctuations, John Wiley &

Sons, New York, 1971.Onsager, L., Phys. Rev., 31, 405, 2265 (1931).Prigogine, I., Introduction to Thermodynamics of Irreversible Processes, John Wiley

& Sons, New York, 1967.Yao, Y. L., Irreversible Thermodynamics, Science Press, Beijing and Van Nostrand-

Reinhold, New York, 1981.

2.2 Common Properties of the Fluxes of Thermodynamic Quantities

Consider a linear tube with cross-section A and length /, filled with amaterial, and assume that changes in the considered thermodynamicquantity can occur only along the length of the tube (i.e. the 'mantle' ofthe tube is insulated, the ends being permeable for the thermodynamicquantities). Flux / is a function of coordinate x (see Fig. 2.1).

82

x-0 x x-/

J(x+Ax)

iFig. 2.1 Balance of fluxes in a linear system

The tube contains the thermodynamic quantity in an amount M (amountof a substance, thermal energy, etc.), which has a density (concentration,energy density, etc.) ip(x) at each point in the tube defined by therelationship

where V is the volume of the system. It thus holds for M that

M = A I ydx (2.2.2)Jo

The change in amount M in the whole tube per unit time is given by therelationship

f 4 fdc [/(/W(0)M (2.2.3)at Jo at

Consider a section of the tube with length Ax; inside this section and inits vicinity, J is a continuous function of x. If an amount J(x)AAt flows intothis section at point x in time At and an amount J(x + Ax)A At at pointx + Ax, then its change in volume A Ax is given as

rx + Ax 3 ^A — dxAt = -A[J(x + AJC) - J(x)]At (2.2.4)

Jx dt

Dividing both sides of the equation by A Ax At and taking the limit yieldsthe continuity relation

The general case of a three-dimensional body enclosed by a surface S will betreated using vector analysis. The time change in the amount M is given bythe relationship

— = -<j)JdS (2.2.6)

<|>JdS=[di\

83

The normal vector dS points out of the body and J > 0 for a quantity flowingout of the body; thus the right-hand side of Eq. (2.2.2) must be negative insign. Obviously,

3M (dip= —dV (2.2.7)

dt J dt V

In view of the Gauss-Ostrogradsky theorem

divJdV (2.2.8)

substitution into Eq. (2.2.6) gives

\^-dV = - fdivJdK (2.2.9)J at j

and differentiation with respect to V yields the continuity equation

- ^ = - d i v J (2.2.10)dt

If in the system the thermodynamic quantity is formed (e.g. by chemicalreaction) at a rate of pi units of quantity per unit volume and unit time, thenEq. (2.2.10) must be completed to yield

-^- = -d iv J + pi (2.2.11)

For thermodynamic quantities other than the amount of substance, thequantity ip can be expressed by the equation

dM dMdw(2.2.12)

where m is the amount of the quantity M per unit of mass (e.g. energy perkilogram), w denotes mass and p is the density of the substance present inthe system. Equation (2.2.11) then takes the form

^ /i (2.2.13)

References

De Groot, S. R., and P. Mazur, see page 81.Denbigh, K. G., see page 81.Ibl, N., Fundamentals of transport phenomena in electrolytic systems, CTE, 6, 1

(1983).Levich, V. G., Physico-Chemical Hydrodynamics, Prentice-Hall, Englewood Cliffs,

1962.

84

2.3 Production of Entropy, the Driving Forces of Transport Phenomena

If the system considered is closed and thermally insulated, then itsentropy increases during a transport phenomenon

f>0 (2.3.1)

If the system is neither closed nor thermally insulated, then the change inthe entropy with time consists of two quantities: of the time change in theentropy as a result of processes occurring within the system St and ofentropy changes in the surroundings, caused by transfer of the entropy fromthe system in the reversible process S"e

£--£ + £ (2.3.2)dt dt dt v }

If we consider a system of infinitesimal dimensions, then the entropybalance can be described directly by Eq. (2.2.13):

p ^ = - d i v J 5 + a (2.3.3)at

where s is the entropy of the system, related to 1 kg of substance, Js is theentropy flux and o is the rate of entropy production per unit volume. Thefirst term on the right-hand side of this equation corresponds to — dSJdt andthe second to dSJdt in Eq. (2.3.2). The rate of entropy formation for asingle transport process (Eq. 2.1.1) is given as

o = ~3X = LX2 (2.3.4)

(unit J m ' V 1 ^ 1 ) . X is the absolute value of the vector of the drivingforce X. Generally,

o = 2 J,X; = 2 2 UXiX* (2.3.5)

Some authors use the dissipation function <l> (unit J m~3 s"1):

< D = - 7 a = - 2 2 LikX,Xk (2.3.6)i k

The rate of entropy production is always positive in the present case,since transport processes are irreversible in nature, i.e. always connectedwith irreversible losses (dissipation) of energy.

The simplest case of a flux and a driving force is shown in the conductionof electric current. This process is governed by Ohm's law for the currentdensity:

(2.3.7)

85

where K is the conductivity and (f> is the electric potential, or for a systemwith a single coordinate x,

dd)j=~Klhc ( Z 3* 8 )

In a unit volume, the passage of current produces heat:

^ (2.3.9)

the rate of entropy production having always a positive value:

f (2.3.10)

In addition to the transport of charge, the current flow in an electrolyte isalso accompanied by mass transport. The migration flux of species i is givenby the equation

J/,migr = -u&FCi grad 0 (2.3.11)

where ut is the mobility^ of the particle (the velocity of the particle underthe influence of unit force, in units of NT1 • mol • m • s"1), zt its chargenumber and c, its concentration. For the migration flux, the driving force isthe molar electric energy gradient multiplied by —1, that is

X m i g r =-z ,Fgrad0 (2.3.12)

The corresponding phenomenological coefficient is then given by therelationship

Lmigr = W/-c, (2.3.13)

The flux of charge, connected with the mass flux of the electricallycharged species, is given by Faraday's law for the equivalence of the currentdensity and the material fluxes:

j = S^J/ (2.3.14)

Thus, in the case of migration material fluxes (Eq. 2.3.11) it holds for thepartial current density (the contribution of the ith ion to the overall currentdensity) that

j , = -UiZJF2^rgrad 0

(2.3.15)

tThe term mobility is used to describe the influence of the drag of the medium on themovement of a particle caused by an unspecified force (the unit of w, is, for example,mol. m s ^ N " 1 ) which may be diffusivity (diffusion coefficient) with a chemical potentialgradient as the driving force and electrolytic mobility connected with the electric field.

86

where Ut is the electrolytic mobility of the ith ion

U^WFut (2.3.16)

The total current density is

J = 2^i h== 2J it J/>migr

= - X Ut \zi\Fc, grad <p = -K grad <j> (2.3.17)I

where K is the conductivity of the system, i.e. this result is identical with Eq.(2.3.7).

Fick's first law,

J, , d i f f=-A grade, (2.3.18)or

//,diff=-A^ (2.3.19)

is an empirical relationship for diffusion. Here, Dt is the diffusion coefficientof the ith component of the system. The concentration gradient cannot beused as a driving force for formulation of the rate of entropy productionaccording to Eq. (2.3.5). Similarly to migration, the gradient of the partialmolar Gibbs energy (chemical potential) multiplied by —1 is selected as thedriving force. Equation (2.3.13) is again used for the phenomenologicalcoefficient. The same value of ut can be used for migration and diffusiononly in dilute solutions. We set

J«,diff = -UiCi grad & = -RTut grad c, (2.3.20)

Comparison with Eq. (2.3.18) yields the relationship for the diffusioncoefficient

D^RTUi (2.3.21)

The ratio of the diffusion coefficient and the electrolytic mobility is given bythe Nernst-Einstein equation (valid for dilute solutions)

Dt RT

The convection flux along the x coordinate,

Ji = civx (2.3.23)

is the amount of substance contained in a column of height vx (the velocityof the medium) and unit base. The general form of Eq. (2.3.23) is

J«.conv = Cl-V (2 .3 .24)

For the sake of completeness, the equation for the heat conduction (the

87

Fourier equation) can be introduced:

J t h = - A g r a d r (2.3.25)

expressing the dependence of the heat energy flux (J • cm"2 • s"1) as afunction of the thermal conductivity A and temperature gradient.

Finally, the scalar flux of the chemical reaction is

Jch = =LA (2.3.26)

where § is the extent of reaction, the phenomenological coefficient L is afunction of the rate constant of the reaction and concentrations of thereactants and A is the affinity of the reaction. The affinity of the chemicalreaction A is the driving force for the chemical flux.

References

See page 81.

2.4 Conduction of Electricity in Electrolytes

2.4.1 Classification of conductors

According to the nomenclature introduced by Faraday, two basic types ofconductors can be distinguished, called first and second class conductors.According to contemporary concepts, electrons carry the electric current infirst class conductors; in conductors of the second class, electric current iscarried by ions. (The species carrying the charge in a given system are calledcharge carriers.)

The properties of electronic conductors follow from the band theory ofthe solid state. The energy levels of isolated atoms have definite values andelectrons fill these levels according to the laws of quantum mechanics.However, when atoms approach one another, their electron shells interactand the positions of the individual energy levels change. When the set ofatoms finally forms a crystal lattice, the original energy levels combine toform energy bands; each of these bands corresponds to an energy level inthe isolated atom. Each energy level in a given band can contain amaximum of two electrons.

When some of the energy bands at a given temperature are completelyoccupied by electrons and bands with higher energy are empty, and alarge amount of energy is required to transfer an electron from the highestoccupied energy band to the lowest unoccupied band, the substance is aninsulator.

Metals are examples of the opposite type of solid substance. Theconductivity band, corresponding to the highest, partially occupied energylevels of the metal atom in the ground state, contains a sufficient number of

Fig. 2.2 Band structure of a semiconductor. eg de-notes the energy gap (width of the forbidden band)

electrons even at absolute zero of temperature to produce the typical highconductivity of metals. These loosely bound electrons form an 'electrongas'. Under the influence of an external electric field, their originallyrandom motion becomes oriented. It should be pointed out that the increasein the electron velocity in the direction of the electric field is much smallerthan the average velocity of their random motion.

Semiconductors form a special group of electronic conductors. These aresubstances with chemically bonded valence electrons (forming a 'valence'band); however, when energy is supplied externally (e.g. by irradiation),these electrons can be excited to an energetically higher conduction band(see Fig. 2.2). A 'forbidden' band lies between the valence and conductivitybands. The energy difference between the lowest energy level of theconductivity band and the highest level of the valence band is termed theband gap sg (the width of the forbidden band).

Electric current is conducted either by these excited electrons in theconduction band or by 'holes' remaining in place of excited electrons in theoriginal valence energy band. These holes have a positive effective charge.If an electron from a neighbouring atom jumps over into a free site (hole),then this process is equivalent to movement of the hole in the oppositedirection. In the valence band, the electric current is thus conducted bythese positive charge carriers. Semiconductors are divided into intrinsicsemiconductors, where electrons are thermally excited to the conductionband, and semiconductors with intentionally introduced impurities, calleddoped semiconductors, where the traces of impurities account for most ofthe conductivity.

If the impurity is an electron donor (for germanium and silicon, group Velements, e.g. arsenic or antimony), then a new energy level is formedbelow the conduction band (see Fig. 2.3A). The energy difference betweenthe lowest level of the conductivity band and this new level is denoted as sd.This quantity is much smaller than the energy gap ed. Thus electrons fromthis donor band pass readily into the conductivity band and represent themain contribution to the semiconductor conductivity. As the charge carriersare electrons, i.e. negatively charged species, these materials are termedn-type semiconductors.

89

excess negative charge

Fig. 2.3 Schematic structure of a silicon semiconductor of (A) n type and (B) /?type together with the energy level array. (According to C. Kittel)

If, on the other hand, the impurity is an electron acceptor (here, elementsof group III), then the new acceptor band lies closely above the valenceband (see Fig. 2.3B). The electrons from the valence band pass readily intothis new band and leave holes behind. These holes are the main chargecarriers in p-type semiconductors.

Compared with metals, semiconductors have quite high resistivity, asconduction of current requires a supply of activation energy. The conduc-tivity of semiconductors increases with increasing temperature.

Germanium provides a good illustration of conductivity conditions insemiconductors. In pure germanium, the concentration of charge carriers isrte = 2 x 1013 cm"3 (in metals, the concentration of free electrons is of theorder of 1022cm~3). The intrinsic conductivity of germanium is about thesame as that of pure (conductivity) water, K = 5.5 x 108 Q~l cm"1 at 25°C.In strongly doped semiconductors, the concentration of charge carriers canincrease by up to four orders of magnitude, but is nonetheless stillcomparable with the concentration of dilute electrolytes.

Ionic (electrolytic) conduction of electric current is exhibited by elec-trolyte solutions, melts, solid electrolytes, colloidal systems and ionizedgases. Their conductivity is small compared to that of metal conductors andincreases with increasing temperature, as the resistance of a viscous mediumacts against ion movement and decreases with increasing temperature.

90

A special class of conductors are ionically and electronically conductingpolymers (Sections 2.6.4 and 5.5.5).

2.4.2 Conductivity of electrolytes

This part will be concerned with the properties of electrolytes (liquid orsolid) under ordinary laboratory conditions (i.e. in the absence of strongexternal electric fields). The electroneutrality condition (Eq. 1.1.1) holdswith sufficient accuracy for current flow under these conditions:

2^, = 0 (2.4.1)i

where ct are the concentrations of ions / and z, are their charge numbers.The basic equations for the current density (Faraday's law), electrolyticmobility and conductivity are (2.3.14), (2.3.15) and (2.3.17).

The conductivity

K = z2F2uiCi = 2 N FUiCi (2.4.2)/ i

in dilute solutions is thus a linear function of the concentrations of thecomponents and the proportionality constants are termed the (individual)ionic conductivities:

li = zJP2ui = \zi\FUi (2.4.3)

Consider a solution in which a single strong electrolyte of a concentrationc is dissolved; this electrolyte consists of v+ cations B2+ in concentration c+

and v_ anions Az~ in concentration c__. Obviously

v+z+ = -v_z_ = v_ |z_| (2.4.4)

and

c. c_c = — = — (2.4.5)

Substitution into Eq. (2.4.2) yields

K = z2+F2u+v+c + Z2-F2u_v_c

= (U+ + U_)z+v+Fc = (U+ + UJ) |z_| v_Fc (2.4.6)

The quantity

(U+ + U_)z+v+F = (U+ + I/_) |z_| v_F = - = A (2.4.7)c

is called the molar conductivity, which is as shown below, a concentration-dependent quantity except in an ideal solution (in practice at high dilution).

91

For first class conductors, the conductivity is a constant characterizing theability of a given material at a given temperature to conduct electric current.However, for electrolyte solutions, it depends on the concentration and isnot a material constant. Thus the fraction A = K/C is introduced; however, itwill be seen below that the constant characterizing the ability of a givenelectrolyte to conduct electric current in solution is given by the limitingvalue of the molar conductivity at zero concentration. The main unit ofmolar conductivity is Q"1 • m2 • mol"1, corresponding to K in Q"1 • m"1 andc in mol • m~3. However, units of Q"1 • cm2 • mol"1 are often used. If unitsof Q"1 • cm"1 are simultaneously used for K and the usual units ofmol • dm"3 for the concentration, then Eq. (2.4.7) becomes

( 2 A 8 )

When reporting the molar conductivity data, the species whose amount isgiven in moles should be indicated. Often, a fractional molar conductivitycorresponding to one mole of chemical equivalents (called a val) isreported. For example, for sulphuric acid, the concentration c can beexpressed as the 'normality', i.e. the species ^H2SO4 is considered.Obviously, A(H2SO4) = 2A(^H2SO4). Consequently, the concept of the'equivalent conductivity' is often used, defined by the relationship

A* = - ^ - = —^— = (U+ + UJ)F = At + A* (2.4.9)

where A* = A+/z+ and A* = A_/|z_| (cf. Eq. 2.4.3). It follows that, for ourexample of sulphuric acid, A*(H2SO4) = A(^H2SO4) = ^A(H2SO4).

Combination of Eqs (2.3.15) and (2.3.17) yields

j ^ - ^ g r a d t f ) (2.4.10)

where tt is the transport (transference) number; giving the contribution ofthe ith ion to the total conductivity K, that is

j

for a single (binary) electrolyte,

U+ + U. At + A!(2.4.12)

t/_ A*

For a solution of a weak electrolyte of total concentration c, dissociating

92

to form v+ cations and v_ anions and with a degree of dissociation or, wehave, considering Eqs (2.4.6) and (2.4.9),

K = a(U+ + U-)z+v+Fc = <x(U+ + (7_) |z_| v_Fc

A = a(U+ + UJ)z+ v+F = a(U+ + UJ) |z_| v_F (2.4.13)A* = a(U+ + UJ)F

The behaviour of real solutions approaches that of ideal solutions at highdilution. The molar conductivity at limiting dilution, denoted A0, is

A0 = z+v+F(U°+ +U°.) = |z_| V-F(U°+ + U°.) = v+k°+ + v_A°_*o *o (2.4.14)

This equation is valid for both strong and weak electrolytes, as a = 1 at thelimiting dilution. The quantities A?= \zt\ FU® have the significance of ionicconductivities at infinite dilution. The Kohlrausch law of independent ionicconductivities holds for a solution containing an arbitrary number of ionspecies. At limiting dilution, all the ions conduct electric current independ-ently; the total conductivity of the solution is the sum of the contributions ofthe individual ions.

Because of the interionic forces, the conductivity is directly proportionalto the concentration only at low concentrations. At higher concentrations,the conductivity is lower than expected from direct proportionality. Thisdecelerated growth of the conductivity corresponds to a decrease of themolar conductivity. Figure 2.4 gives some examples of the dependence of

0.3

Fig. 2.4 Dependence of molar conductivity of strong elec-trolytes on the square root of concentration c. The dashed

lines demonstrate the Kohlrausch law (Eq. 2.4.15)

93

the molar conductivity of strong electrolytes on the concentration. It can beseen that even at rather low concentrations, the limiting molar conductivityis still not attained.

A special branch of the theory of strong electrolytes deals with thedependence of the electrical conductivity of electrolytes on concentration(see Section 2.4.3). For very low concentrations, Kohlrausch found empiri-cally that

A = A°-Ax1/2 (2.4.15)

(k is an empirical constant), i.e. the dependence of A on cm is linear.Interionic forces are relatively less important for weak electrolytes

because the concentrations of ions are relatively rather low as a result ofincomplete dissociation. Thus, in agreement with the classical (Arrhenius)theory of weak electrolytes, the concentration dependence of the molarconductivity can be attributed approximately to the dependence of thedegree of dissociation a on the concentration. If the degree of dissociation

« ~ ^ (2.4.16)

is substituted into the equation for the apparent dissociation constant K',then the result is

sometimes called the Ostwald dilution law. In a suitably linearized form,this equation can be used to calculate the quantities K' and A0 frommeasured values of A and c. However, the resulting values of K' are notthermodynamic dissociation constants; the latter can be found fromconductivity measurements by a more complicated procedure (see Section2.4.5). Figure 2.5 illustrates, as an example, the dependence of the molarconductivity on the concentration for acetic acid compared with hydrochlo-ric acid.

While the molar conductivity of strong electrolytes A0 can be measureddirectly, for determination of the ionic conductivities the measurabletransport numbers must be used (cf. Eq. (2.4.12)). Table 2.1 lists the valuesof the limiting conductivities of some ions in aqueous solutions.

2.4.3 Interionic forces and conductivity

The influence of interionic fores on ion mobilities is twofold. Theelectrophoretic effect (occurring also in the case of the electrophoreticmotion of charged colloidal particles in an electric field, cf. p. 242) is causedby the simultaneous movement of the ion in the direction of the applied

94

400

I 1 I I

0.1

Fig. 2.5 Dependence ofthe molar conductivity ofthe strong electrolyte (HCl)and of the weak electrolyte(CH3COOH) on the square

root of concentration

Table 2.1 Ionic conductivities at infinite dilution (Q 1 • cm2 • mol *) at varioustemperatures. (According to R. A. Robinson and R. H. Stokes)

Ion

H+

OH"Li+

Na+

K+

Rb+

Cs+

crBr~

r

0

225.0105.019.426.540.743.944.041.042.641.4

5

250.10—

22.7630.3046.7550.1350.0347.5149.2548.57

15

300.60—

30.2039.7759.6663.4463.1661.4163.1562.17

Temperature (°<

18

31517132.842.863.966.56766.068.066.5

25

349.81198.3038.6850.1073.5077.8177.2676.3578.1476.84

^)

35

397.0—

48.0061.5488.2192.9192.1092.2194.0392.39

45

441.4—

58.0473.73

103.49108.55107.53108.92110.68108.64

55

483.1

68.7486.88

119.29124.25123.66126.40127.86125.44

100

630450115145195

—212

—

95

Fig. 2.6 Electrophoretic effect. The ionmoves in the opposite direction to the ionic

atmosphere

electric field and of the ionic atmosphere in the opposite direction (Fig.2.6). Both the central ion and the ions of the ionic atmosphere take theneighbouring solvent molecules with them, which results in a retardation ofthe movement of the central ion.

For very dilute solutions, the motion of the ionic atmosphere in thedirection of the coordinates can be represented by the movement of asphere with a radius equal to the Debye length LU = K~1 (see Eq. 1.3.15)through a medium of viscosity r/ under the influence of an electric forcezteExy where Ex is the electric field strength and zt is the charge of the ionthat the ionic atmosphere surrounds. Under these conditions, the velocity ofthe ionic atmosphere can be expressed in terms of the Stokes' law (2.6.2) bythe equation

(2.4.18)

The electrolytic mobility of the ionic atmosphere around the ith ion canthen be defined by the expression

zte6;rr/LD 6jrNAr]Lr>

(2.4.19)

This quantity can be identified with deceleration of the ion as a result ofthe motion of the ionic atmosphere in the opposite direction, i.e.

(2.4.20)

96

Fig. 2.7 Time-of-relaxation effect. During themovement of the ion the ionic atmosphere isrenewed in a finite time so that the position ofthe ion does not coincide with the centre of the

ionic atmosphere

The time-of-relaxation effect (see Fig. 2.7) originates in a certain timedelay (relaxation) required for the renewal of the spherical symmetry of theionic atmosphere around the central ion moving under the influence of anapplied electric field. The disappearance of the ionic atmosphere afterremoval of the central ion, similar to its formation, is an exponentialfunction of time; in fact, both of these processes are complete after twicethe relaxation time, which is of the order of 10~7 to 10~9 s, depending on theelectrolyte concentration. If the central ion moves under the influence of anexternal electric field, it becomes asymmetrically located with respect to thecentre of the ionic atmosphere. Thus the time average of the forces ofinteraction of the ionic atmosphere with the central ion is not equal to zero.The external electric field is decreased by the relaxation electric field, as it isoriented in the opposite direction to the external force. Although therelaxation time is several orders of magnitude smaller than the timerequired for the central ion to pass through the ionic atmosphere (about10~3s), its effect is important because the strength of the electric fieldformed by the ionic atmosphere (~105 V • cm"1) is greater than the strengthof the external electric field. Thus, even small changes in the symmetry ofthe ionic atmosphere have a measurable effect acting against that of theexternal electric field.

The mathematical theory of the time-of-relaxation effect is based on theinterionic electrostatics and the hydrodynamic equation of flow continuity.It is the most involved part of the theory of strong electrolytes. Only themain conclusions will be given here.

97

The first approximate calculation was carried out by Debye and Hiickeland later by Onsager, who obtained the following relationship for therelative strength of the relaxation field AE/E in a very dilute solution of asingle uni-univalent electrolyte

In the ideal case, the ionic conductivity is given by the productBecause of the electrophoretic effect, the real ionic mobility differs from theideal by ALf, and equals U° + AC/,. Further, in real systems the electric fieldis not given by the external field E alone, but also by the relaxation fieldAE, and thus equals E + AE. Thus the conductivity (related to the unitexternal field E) is increased by the factor (E + AE)/E. Consideration ofboth these effects leads to the following expressions for the equivalent ionicconductivity (cf. Eq. 2.4.9):

(2.4.22)

+

and for the overall equivalent conductivity of the electrolyte

A* = F(U°+ + AU+ + l/L + Al/_)(l + — )

- F[{U°+ + U°_)(l + ^ \ + AU+ + AU_1 (2.4.23)

We shall introduce FU°+ = X°+, FU°- = X0_ and F(U% +l/L) = A°. Theresulting Onsager's expression (2.4.21) completed by the term 1 + BaVc inthe denominator introduced by Falkenhagen (cf. Eq. (1.3.31)) takes botheffects of interionic forces at higher concentrations into account:

V = A'0-(B IA° + 3 2 ) r ^ - (2.4.24)

where

B p2K ^1 12n(2 + y/2)NAeRTVc \2n{\ + V2)(eRT)3'2

2 6JZNAY)VC 3jtNArj(£RT)m

The validity of Eq. (2.4.24) for uni-univalent electrolytes has been verifiedup to a concentration of 0.1 mol • dm"3. If the ionic radius is not involved in

98

Eq. (2.4.24) (i.e. the product BaVc is cancelled), then the Onsager limitinglaw is obtained

A = A0 - (B.A0 + B2)\fc (2.4.26)

which is an analogy of the Debye-Hiickel limiting law. In the same way it isvalid for rather dilute solutions (see Fig. 2.4, dashed line). The ratio of themolar conductivities at moderate and limiting dilutions is the conductivitycoefficient given for rather dilute solutions by the relation

yA = 1 - (fli + B2lA°)Vc (2.4.27)

In an analogous way, for a weak uni-univalent electrolyte the Onsagerlimiting law has the form

A = or[A° - (BiA0 + B2)Vac] (2.4.28)

The ratio of the equivalent conductivity at a given concentration to thelimiting equivalent conductivity then is

A/A°=aryA (2.4.29)

These relations are used in the precise form of Ostwald's dilution law(2.4.17).

Equations (2.4.27) and (2.4.28) are employed for conductometric deter-mination of dissociation constants and solubility products.

2.4.4 The Wien and Debye-Falkenhagen effects

The conductivities of strong electrolytes do not depend on the strength ofthe electric field for weak fields (of the order of 104V-m~1). At highelectric field strengths (of about 107V-m~1), Wien observed a significantincrease in the conductivity (Fig. 2.8). This effect increases at higherconcentrations and at a higher charge number of the electrolyte ions andapproaches a limiting value with increasing electric field. This phenomenonis a result of the high ion velocities, preventing rearrangement of the ionicatmospheres during motion. Thus an ionic atmosphere is not formed at alland both the electrophoretic and relaxation effects disappear.

The conductivity also increases in solutions of weak electrolytes. This'second Wien effect' (or field dissociation effect) is a result of the effect ofthe electric field on the dissociation equilibria in weak electrolytes. Forexample, from a kinetic point of view, the equilibrium between a weak acidHA, its anion A" and the oxonium ion H3O+ has a dynamic character:

HA + H2O <=± H3O+ + A"

where kd is the rate constant for dissociation and kT is the rate constant forrecombination of the anion with a hydronium ion. Their ratio yields the

99

10 20Electric field, 104V.cm1

Fig. 2.8 The Wien effect shown by the percentage increase ofequivalent conductivity in dependence on the electric field inLi3Fe(CN)6 solutions in water. Concentrations in mmol • dm"3 are

indicated at each curve

sociation constant for a weak acid

(2.4.30)

e bond between the hydrogen atom and the anion is primarily electro-tic in character, so that the acid molecule has the character of an ion paira certain extent. As demonstrated by Onsager, the rate of dissociation ofion pair increases in the presence of an external electric field, while the

100

rate of formation of this ion pair is not affected by the presence of anexternal field. The value of kd thus increases in the presence of a field, asdoes KA. A weak electrolyte is therefore dissociated to a greater degree instrong electric fields and its conductivity increases.

Debye and Falkenhagen predicted that the ionic atmosphere would notbe able to adopt an asymmetric configuration corresponding to a movingcentral ion if the ion were oscillating in response to an applied electricalfield and if the frequency of the applied field were comparable to thereciprocal of the relaxation time of the ionic atmosphere. This was found tobe the case at frequencies over 5 MHz where the molar conductivityapproaches a value somewhat higher than A0. This increase of conductivityis caused by the disappearance of the time-of-relaxation effect, while theelectrophoretic effect remains in full force.

2.4.5 Conductometry

Determination of the conductivity of electrolyte solutions. The resistanceof second class conductors is measured by the bridge method, similar tomeasurement of the resistance of first class conductors. Usually, alternatingcurrent is used, to avoid polarization of the electrodes (see Chapter 5). Thisinvolves certain technical difficulties, connected with the necessity ofcompensating the capacity and inductive components of the resistors andalso with the fact that current can pass through the capacity couplingbetween the circuit and ground; similarly, strong magnetic fields produceinduction effects. Direct current can only be used if a suitable unpolarizableelectrode can be found for the given system (see page 209). Therefore, mostoften an a.c. supply of audio-frequency and an amplitude of a few millivoltsis used.

The bridge method is based on various modifications of the well-knownWheatstone bridge.

Instruments for exact measurements usually have a sinusoidal currentsource and an electronic balance detector. The circuit is made as symmetri-cal as possible to avoid stray coupling.

During measurement, the conductivity cell is filled with an electrolytesolution; this cell is usually made of glass with sealed platinum electrodes.Various shapes are used, depending on the purpose that it is to serve.Figure 2.9 depicts examples of suitable cell arrangements. The electrodesare covered with platinum black, to avoid electrode polarization. Theelectrodes are placed close to one another in poorly conductive solutionsand further apart in more conductive solutions.

For an evaluation of measurements the conductivity cell is calibrated witha solution of a known conductivity. Both the electrolyte and the solventmust be carefully purified.

101

A B

Fig. 2.9 Conductometric cells for (A) low and (B) high conductivity solutions

2.4.6 Transport numbers

Transport (transference) numbers are defined by Eqs (2.4.11) and(2.4.12). The experimental arrangement shown in Fig. 2.10 can be used todemonstrate this concept. In the cathode and anode compartments theelectrolyte concentration is maintained homogeneous by stirring. Theconnecting tube is narrow, so that it contains a negligible amount ofelectrolyte compared with that in the electrode compartments. The electricfield is homogeneously distributed and mass transfer occurs only throughmigration at sufficiently high field strengths. If the charge Q passes throughthis electrolysis cell, then Q/z+F moles of cations are discharged on thecathode and Q/\z-\F moles of anions on the anode. An amount ofQtJ\z-\ F moles of anions migrates from the cathode space to the anodespace and Qt+/z+F moles of cations in the opposite direction. Thus thecathode compartment will be depleted by a total of An+>c = Qt_/z+F molesof cations and An_ c = Qf_/|z_| F moles of anions, and the anode compart-ment by An+ A = Qt+/z+F moles of cations and An_ A = Qt+/\z_\ F moles

H \-Fig. 2.10 Schematic design of a cell for the determina-tion of transport numbers from measurements of theconcentration decrease in electrode compartments

(Hittorf s method)

102

of anions. As it obviously holds that n+ = \z_\n and n_ = z+n for theamounts of cations n+, anions n_ and salt n, respectively, then the decreasein the amount of the salt is Anc = Qt_/z+ |z_| F and AnA = Qt+/z+ |z_| F.Thus the decrease in the amount of salt in the cathode compartment isproportional to the transport number of the anion and the decrease in theamount of salt in the anode compartment is proportional to the transportnumber of the cation,

t+ = —A— and f_ = — (2.4.31)Anc + AnA Anc + AnA

Thus, the transport number can be found from measurement of thedecrease in the amount of the salt in the electrode compartments.

In view of Eq. (2.4.9), for a strong electrolyte,

1 * 1 * 3 * 7*

' + = F T T = A ^ and '- = r T r = X ^ (2A32)

and, for a weak electrolyte,

'+ = 7 ^ and '- = f r (2-4-33)These equations are used to determine ionic conductivities.

The transport numbers thus depend on the mobilities of both the ions ofthe electrolyte (or of all the ions present). This quantity is therefore not acharacteristic of an isolated ion, but of an ion in a given electrolyte. Table2.2 lists examples of transport numbers. It can be seen from the table thatthe transport numbers also depend on the electrolyte concentration. Thefollowing rules can be derived from experimental data:

(a) If the transport number is close to 0.5, it depends only very slightly onthe concentration.

(b) If the transport number for the cation is less than 0.5, then it decreaseswith increasing concentration (and, simultaneously, t_>0.5 increaseswith increasing concentration); and vice versa.

These rules are based on the theory of conductivity of strong electrolytesaccounting for the electrophoretic effect only (the relaxation effect termsoutbalance each other).

The methods for determination of transport numbers include the Hittorfmethod and the concentration cell method (p. 121).

The Hittorf method is based on measuring the concentration changes atthe anode and cathode during electrolysis. These changes can be found by asensitive analytical method, e.g. conductometrically for a suitable cell

103

Table 2.2 Transport numbers of cations at various concentrations(mol • dm"3). Relative accuracy 0.02 per cent. (According to B. E.

Conway)

Electrolyte

HC1CH.COONaCH3COOKKNO3NH4CIKC1KIKBrAgNO3

NaClLiClCaCl2

|Na2SO4

±K2SO4

^LaCl3|K4Fe(CN)6

3-K3Fe(CN)6

0

0.82090.55070.64270.50720.49090.49060.48920.48490.46430.39630.33640.43800.3860.4790.477

——

0.01

0.82510.55370.64980.50840.49070.49020.48840.48330.46480.39180.32890.42640.38480.48290.46250.515

—

c

0.02

0.82660.55500.65230.50870.49060.49010.48830.48320.46520.39020.32610.42200.38360.48480.45760.555

—

0.05

0.82920.55730.65690.50930.49050.48990.48820.48310.46640.38760.32110.41400.38290.48700.44820.6040.475

0.10

0.83140.55940.66090.51030.49070.48980.48830.48330.46820.38540.31680.40600.38280.48900.43750.6470.491

0.20

0.83370.5610

—0.51200.49110.48940.48870.4841

—0.38210.31120.39530.38280.49100.4233

——

arrangement. Transport numbers can be found by using Eq. (2.4.31) fromthe overall change in electrolyte contents in the anode and cathodecompartments.

Correct transport numbers could be obtained from concentration changesin the electrode compartments only if the ions were not hydrated in theaqueous solution. When they are hydrated the accompanying watermolecules remain in the electrode compartment during discharge of theions. Accordingly, the measured decrease in electrolyte concentration at theelectrode is greater (or smaller) than would correspond to simple chargetransport. If n+ molecules of water are bound to each cation n_ and to eachanion, then W moles of water are transported by a charge passagecorresponding to 1 F:

W = n+t+ - n_t_ (2.4.34)

If the cation is more hydrated, then W is a positive number; if the anion ismore hydrated, then W is a negative number and water is transported to theanode. Transport numbers calculated from measured concentration changesinvolving transport of water by solvated ions are sometimes called Hittorf(//) numbers; those corrected for the transport of water are called truetransport numbers (*,-). These two types of transport numbers are related by

104

the following expressions:

cW rWt+ t + + a n d ' - = ' ! - * , (2.4.35)

55.5z+v+ 55.5 |z_| v_If water is transported to the cathode, then W>0, t+>t+ and *_<*!_. Ifwater is transported to the anode, then W <0, t+<t'+ and t_>t'_. Ionsolvation can be determined from measured true and Hittorf transportnumbers. The value of W can be found experimentally by adding anon-electrolyte (e.g. sucrose) to the solution. As it is assumed that thissubstance does not migrate in the electric field, W is determined bymeasuring changes in its concentration in the electrode compartment afterelectrolysis. It is evident from Eq. (2.4.34) that, even when the truetransport numbers and the values of W are known, it is not possible todetermine the absolute values of hydration numbers, but only relativevalues under certain assumptions (e.g. that the bromide ion is nothydrated).

References

Barthel, J., Transport properties of electrolytes from infinite dilution to saturation,PureAppl. Chem., 57, 355 (1985).

Falkenhagen, H., Theorie der Elektrolyte, Hirzel, Leipzig, 1971.Fuoss, R. M., and F. Accascina, Electrolytic Conductance, Interscience, New York,

1959.Justice, J.-C, Conductance of electrolyte solutions, CTE, 5, Chap. 3 (1982).Kittel, C , Introduction to Solid State Physics, John Wiley & Sons, New York, 1976.Robbins, J., Ions in Solution, Vol. 2, An Introduction to Electrochemistry, Oxford

University Press, Oxford, 1972.Shedlovsky, T., and L. Shedlovsky, Conductometry, in Physical Methods of

Chemistry (Eds. A. Weissberger and B. W. Rossiter), Part IIA, Wiley-Interscience, New York, 1971, p. 163.

2.5 Diffusion and Migration in Electrolyte Solutions

The passage of current through electrolytes as treated in Section 2.4 didnot involve any transient phenomena. During migration the current and thepotential difference were assumed constant. During the measurement,transient phenomena appear only in connection with electrode charging orwith reactions at the electrodes from which electric charge is transferredinto the electrolyte. One of the aims of experimental methods is toeliminate this effect during measurements.

On the other hand, it is characteristic for diffusion processes that theyusually involve time changes of the concentrations of the diffusing sub-stances, so that they have a transient character. Steady state is a result of aprior transient phenomenon that is inherent in the diffusion process itself.

105

It is characteristic for the actual diffusion in electrolyte solutions that theindividual species are not transported independently. The diffusion of thefaster ions forms an electric field that accelerates the diffusion of the slowerions, so that the electroneutrality condition is practically maintained insolution. Diffusion in a two-component solution is relatively simple (i.e.diffusion of a binary salt—see Section 2.5.4). In contrast, diffusion in athree-component electrolyte solution is quite complicated and requires theuse of equations such as (2.1.2), taking into account that the flux of oneelectrically charged component affects the others.

A simple case is the diffusion of a single type of ion in a solutioncontaining a sufficient excess of an indifferent electrolyte (see page 116),which then occurs in the same way as in the case of a non-electrolyte.Isotope (tracer) diffusion has the same character, where a concentrationgradient of the radioactive isotope of an ion, present in a much lowerconcentration, is formed in a solution with a much larger, constant saltconcentration.

2.5.1 The time dependence of diffusion

We will first consider the simple case of diffusion of a non-electrolyte.The course of the diffusion (i.e. the dependence of the concentration of thediffusing substance on time and spatial coordinates) cannot be deriveddirectly from Eq. (2.3.18) or Eq. (2.3.19); it is necessary to obtain adifferential equation where the dependent variable is the concentration cwhile the time and the spatial coordinates are independent variables. Thederivation is thus based on Eq. (2.2.10) or Eq. (2.2.5), where we set ip = cand substitute from Eq. (2.3.18) or Eq. (2.3.19) for the fluxes. This yieldsFick's second law (in fact, this is only a consequence of Fick's first lawrespecting the material balance—Eq. 2.2.10), which has the form of apartial differential equation

^ = D div grad c = DV2c (2.5.1)at

where V2 is the Laplace operator (for its form in Cartesian and sphericalcoordinates, see footnote p. 109). When the concentration is a function ofcoordinate x alone, Fick's first law has the most common form:

and the second law is then

In order to integrate the partial differential equations (2.5.1) or (2.5.3) it

106

is necessary to choose suitable initial and boundary conditions, i.e. to definea model of a certain experiment. The initial condition describes theconcentration conditions of the system at the beginning of the diffusionprocess. The values of the concentrations or material fluxes (which areproportional to concentration gradients) at the boundary surfaces of thesystem or at concentration discontinuities inside the system are described bythe boundary conditions.

For illustration, consider the simplest type of diffusion, described by thepartial differential equation (2.5.3), also called linear diffusion. The systemwill be represented by an infinite tube closed at one end (for x = 0) andinitially filled with a solution with concentration c°. Diffusion is produced byvery fast removal (e.g. by precipitation or an electrode reaction) of thedissolved substance at the x = 0 plane (the reference plane). The initialconcentration c° is retained at large distances from this reference plane(x—>oo). The initial condition is thus

x > 0, t = 0,

and the boundary conditions are

JC = O, t>0,

c = c° (2.5.4)

c = 0

c = c° (2.5.5)

Solution of the differential equation (2.5.3) together with the initial andboundary conditions (2.5.4) and (2.5.5) yields the relationship

(2.5.6)

where the error function erf y is defined by the equation

erf y =

Table 2.3 Error function erf (x) = (2/VJT) Jj e"y2 dy

X

00.10.20.30.40.50.60.70.80.91.0

erf(jc)

0.0000.1130.2230.3290.4280.5210.6040.6780.7420.7970.843

X

1.11.21.31.41.51.61.71.81.92.0

erf(x)

0.8800.9100.9140.9520.9660.9760.9840.9890.9930.995

For x < 0.1 in sufficient approximation erf (x) = 1. 128JC - 0.376*

107

0.05Distance! cm

Fig. 2.11 Concentration distribution in the case of linear diffusion withc = 0 for x = 0 (see Eq. 2.5.6), where D = 10"' • s , time in secondsis indicated at each curve and the effective diffusion layer thickness is

shown for t = 100 s

(see Table 2.3). Figure 2.11 depicts the concentration distribution. It can beseen from Eq. (2.5.6) that the ratio c/c° is independent of the concentrationbut only depends on the distance from the reference plane, on the time andon the diffusion coefficient. Assuming that the solution of the diffusingsubstance is dilute, the diffusion coefficient does not depend on itsconcentration.

The material flux through the reference plane is given by the concentra-tion gradient for x = 0. In the present case the concentration gradient is

dc

dx'~ {nDtf 4Dt)

and the concentration gradient at the reference plane is then

(2.5.7)

(2.5.8)

Obviously, the material flux decreases with increasing time and, finally, fort—><*> approaches zero.

If a tangent to the curves in Fig. 2.11 is drawn at the origin, it intersectsthe straight line c = c° at a distance of

6D = (nDt)1/2 (2.5.9)

108

termed the diffusion layer thickness. It is a measure of the region that isdepleted by the diffusion process.

Another example is linear diffusion, with a prescribed concentrationgradient at the reference plane, i.e. a prescribed material flux through thereference plane. This type of diffusion transport is important mainly forelectrode processes (see Section 5.4). The point of interest in this case is theconcentration at the reference plane. In the simplest case, the material fluxis constant, so that the boundary condition for x=0 (Eq. 2.5.5) can bereplaced by

= 0, t>0, D^ =dx

(2.5.10)

The resultant concentration distribution (Fig. 2.12) is given by the equation

2 \ K / x

(

The error function complement erfcy is defined by the relationshiperfcy = 1 — erf y. The concentration at the reference plane c* is

c* = c°- 2Ktm(jzD)-m (2.5.12)

Clearly, the concentration at the reference plane decreases to zero after

0.002 0.004

Distance, cm0.006 0.008

Fig. 2.12 Concentration distribution for linear diffusion and constantconcentration gradient at the reference plane x = 0 (Eq. 2.5.11), where- - • - " - - ' D = 1 0 - 5 c m 2 - s - \ c° = 5 x 10~5 mol • cnT3

(T = 1.83 s)

109

time

c°\2

This case is an example of a time-restricted process as it cannot proceedafter the instant t=x (the concentration would assume negative values).Time r is termed the transition time.

Consider a tube of length / containing a solution of concentration c2, withone end (x = 0) rinsed with a solution of concentration cx and the other(JC = /) with a solution of concentration c2. In contrast to the previousexample, after a period of time steady state will be reached, characterized bythe relationship

f = 0 (2.5.14)

for f—»oo. The boundary conditions for the ordinary differential equation

d2cD - 5 = 0 (2.5.15)

are

' C~Cl (2.5.16)X = / , C = C2

The solution of this system is then

x x(2.5.17)

The material flux at the steady state 7st is constant at every point inthe system:

^ ^ (2.5.18)

As a last example in this section, let us consider a sphere situated in asolution extending to infinity in all directions. If the concentration at thesurface of the sphere is maintained constant (for example c = 0) while theinitial concentration of the solution is different (for example c = c°), thenthis represents a model of spherical diffusion. It is preferable to express theLaplace operator in the diffusion equation (2.5.1) in spherical coordinatesfor the centro-symmetrical case.t The resulting partial differential equation

2) + (d2/dz2) = (d2/dr2) + (d/dr)(2/r), where r is the distance fromthe origin (located at the centre of the sphere).

dc (d2c 2 3c

110

is then

r>r0, t = 0, c = c°

r = r0, t>0, c = 0 (2.5.19)

r^oo, t>0, c = c°

where r0 is the radius of the sphere.The concentration gradient for r = r0 is given by the equation

(dc\ c° c°— = - 7 = ^ + - (2.5.20)

The concentration gradient will obviously assume the steady-state value fort»rl/nDy

dr/r=ro r0

Thus the time during which the transport process attains the steady statedepends strongly on the radius of the sphere r0. The steady state isconnected with the dimensions of the surface to which diffusion transporttakes place and does, in fact, not depend much on its shape. Diffusion to asemispherical surface located on an impermeable planar surface occurs inthe same way as to a spherical surface in infinite space. The properties ofdiffusion to a disk-shaped surface located in an impermeable plane are notvery different. The material flux is inversely proportional to the radius ofthe surface and the time during which stationary concentration distributionis attained decreases with the square of the disk radius. This is especiallyimportant for application of microelectrodes (see page 292).

Further examples of diffusion processes characterized by boundaryconditions connected with specific electrode processes will be considered inSection 5.4.

2.5.2 Simultaneous diffusion and migration

Consider a dilute electrolyte solution containing s components (non-electrolytes and various ionic species) in which concentration gradients ofthe components and an electric field are present. The material flux of the ithcomponent is then given by a combination of Eqs (2.3.11) to (2.3.20):

J, = -ut(RT grad ct + c^F grad <j>) (2.5.22)

This relationship can be expressed in the form

J,-=-!*,•<:,• grad ft (2.5.23)

I l l

where the driving force for the simultaneous diffusion and migration is thegradient of the electrochemical potential /i, = jU, + z{F(^. The concept of theelectrochemical potential is discussed in detail in Section 3.1.1.

Equations (2.3.11), (2.3.16) and (2.3.18) can be combined to give

J, = -Dt grad c, - - ^ UiCi grad 0 (2.5.24)

or, for transport with respect to coordinate x alone,

h - - A -* :—r UiCj —djc \zt\ dx

dc d(t>= -RTu,--- ZiFutCi-^ (2.5.25)

djc dx

Equations (2.5.22) and (2.5.25) are called the Nernst-Planck equations.Substitution for J, from Eq. (2.5.24) into Faraday's law (2.3.14) yields the

equation

j = - S ZiFDt grad c, - 2 \zt\ FU& grad 0 (2.5.26)

which, combined with the conductivity relationship (2.4.2), yields

j = - 2 ZiFDi grad ct - K grad </> (2.5.27)

The electric potential gradient is thus given by the sum of twoexpressions:

—j I s

grad <t> = 2 ZiFDf grad c,K K i=1

= grad 0ohm + grad 0diff (2.5.28)

The first of these is the ohmic potential gradient, characteristic for chargetransfer in an arbitrary medium. It is formed only when an electric currentpasses through the medium. The second expression is that for the diffusionpotential gradient, formed when various charged species in the electrolytehave different mobilities. If their mobilities were identical, the diffusionelectric potential would not be formed. In contrast to the ohmic electricpotential, the diffusion electric potential does not depend directly on thepassage of electric current through the electrolyte (it does not disappear inthe absence of current flow).

2.5.3 The diffusion potential and the liquid junction potential

If all the changes taking place in the system occur along the direction ofthe x axis, then Eqs (2.5.28), (2.4.2), and (2.4.11) yield a relationship for

112

the diffusion potential gradient

A A, D T S Z'U' d C i / 6 X DT ^ Z ' " ' C ' d m C< / d*d0 Ki i = l A l i = l

_ RT* tt d In cf (2.5.29)

The electric potential difference between two points (2 and 1) in theelectrolyte during diffusion is thus given by the equation

D T /"2 S *

diff =(p2-<t>l= -—\ E -dInc7A0diff =(p2-<t>l= -—\ E -dInc 7 (2.5.30)

where the integration limits refer to the solution composition at points 1 and2. In general the value of A$diff is influenced by the way in which the c/sand ff-'s depend on x. The transport number for a dilute solution is not afunction of the concentration only in the case of a single binary salt (see Eqs2.4.12), so that in this case the diffusion potential is given by the equation

z+-(z+-z.)t+RT c(l)A0diff = — — I n — (2.5.31)

where c(l) and c(2) are the concentrations at points 1 and 2.The potential difference formed at the junction of two electrolytes with

different but constant composition and dissolved in the same solvent is veryimportant in electrochemical measurements. Solutions can come into contacteither directly (a suitable experimental arrangement must be used toprevent mixing through convection—see below) or can be separated by adiaphragm with sufficiently large pores. This diaphragm can be a glass frit,ceramic disk, etc., and is permeable for all the components of the systemand simply prevents mechanical mixing. Thus, the system consists of twosolutions, where the concentration is constant everywhere, and of atransition region at their contact, where diffusion occurs. This region has athickness of d and is either free or enclosed in the pores of a diaphragm:

Solution 11 diaphragm | solution 2x = p x = q

Here, x is the coordinate normal to the diaphragm, so that d = q —p. Theliquid junction potential A0L is the diffusion potential difference betweensolutions 2 and 1. The liquid junction potential can be calculated for morecomplex systems than that leading to Eq. (2.5.31) by several methods. Ageneral calculation of the integral in Eq. (2.5.30) is not possible and thusassumptions must be made for the dependence of the ion concentration on xin the liquid junction. The approximate calculation of L. J. Henderson is

113

often used, assuming that the ion concentration in the liquid junction varieslinearly with x between the values in the two solutions:

ct(x) = Ci(p) + [cXq) — C;(p)] — (2.5.33)

Substitution into Eq. (2.5.30) yields

RT Uu^c^-cApWd + c^p)}

Y.zjuj{[Cj(q)-cJ(p)]x/d + Ci7 = 1

1 = 1

D _ E ZiUtlCiiq) ~ Ci(p)] E zjutCii

T ?E Z^M^CK?)-^) ] E z ? ^

/=i 1=1

whre d = q —p is the thickness of the liquid junction.Although the Henderson formula depends on a number of simplifications

and also employs ion concentrations rather than activities, it can nonethe-less be used for estimation of the liquid junction potential up to moderateelectrolyte concentrations, apparently as a result of compensation of errors.

If there is only a single uni-univalent electrolyte with a common anion orcation on both sides of the liquid junction and the concentration of bothelectrolytes is the same, e.g.

O . I M K C I

solution 1

O . I M H C I

solution 2

then Eq. (2.5.34) is reduced to the form (Lewis-Sargent equation)

RT U+{2) + V-{2) RT A(2)=± V = ±1"where the symbols 1 and 2 designate the electrolytes in solutions 1 and 2and the A's are the molar conductivities at the given electrolyte concentra-tions. The minus sign holds when the electrolytes have a common anion andthe plus sign holds for a common cation. The Henderson equation for aconstantly varying electrolyte mixture approximately approaches conditionsfor a liquid junction with free diffusion (see Fig. 2.13).

The general Planck's solution of the liquid-junction potential is based onthe assumption that the fluxes of ions in the liquid junction are in a steadystate. As the mathematical treatment is rather space-consuming the readeris recommended to inspect a more recent treatment of this by W. E. Morf

114

to sample to referencesolution ^ ^ _ ^ L ^ e l e c t r o d e P i x

bridge

to waste \ c e r a m i c plug

A BFig. 2.13 Examples of liquid junctions: (A) liquid junctionwith 'free' diffusion is formed in a three-way cock whichconnects the solution under investigation with the salt bridgesolution; (B) liquid junction with restrained diffusion isformed in a ceramic plug which connects the salt bridge with

the investigated solution

(Anal. Chem.y 49, 810, 1977). Planck's model corresponds to the liquidjunction formed inside rather thin-pore diaphragms (Fig. 2.13B).

However, useful arrangements are made to practically eliminate thediffusion potential, i.e. reduce it to the smallest possible value, inpotentiometric measurements not concerned directly with measuring theliquid junction potential, e.g. pH measurements, potentiometric titrations,etc. This is achieved by using a salt bridge connecting the two solutions,consisting, for example, of a tube closed at both ends by a porous materialand usually filled with a saturated solution of potassium chloride, am-monium nitrate, or sodium formate (if the presence of chloride orpotassium ions is undesirable). The theoretical basis for the use of these saltbridges is that the ions of the bridge solutions are present in a large excesscompared with the ions in the electrode compartments, and thus are almostsolely responsible for the charge transport across the two boundaries. Asthe transport numbers of the cations and anions of these excess electrolytesare almost identical, the two newly formed diffusion potentials are small andact in opposite directions, so that the final diffusion potential is negligible.For example, the diffusion potential at the contact between 0.1 and0.01 M HC1 is about 40 mV. If a salt bridge containing a saturated potassiumchloride solution is placed between the two solutions, the diffusionpotential is about 3 mV on the side with the more dilute solution and about5 mV on the side with the more concentrated solution. The final potentialdifference is about 2 mV, which is insignificant in many applications.

In potentiometric measurements the simplest approach to the liquid-junction problem is to use a reference electrode containing a saturatedsolution of potassium chloride, for example the saturated calomel electrode(p. 177). The effect of the diffusion potential is completely suppressed if thesolutions in contact contain the same indifferent electrolyte in a sufficient

115

excess, which increases the overall conductivity of the solutions (cf. Eq.2.5.28).

2.5.4 The diffusion coefficient in electrolyte solutions

As demonstrated in the preceding section, an electric potential gradient isformed in electrolyte solutions as a result of diffusion alone. Let us assumethat no electric current passes through the solution and convection is absent.The Nernst-Planck equation (2.5.24) then has the form:

J, = - A grad c, - - ^ UiCi grad 0dif (2.5.36)

Substitution from Eq. (2.3.22) into Eq. (2.5.36) yields the followingrelationship for dilute solutions:

D zFJt = - A grad c, - - ^ - Ci grad 0dif (2.5.37)

For a single electrolyte, Eqs (2.5.37) and (2.3.14) for y = 0 and Eqs(2.4.4) and (2.4.5) yield the equation

where D+ and D_ are the diffusion coefficients of the cation and anion.The overall material flux of the electrolyte

J = — = — (2.5.39)v+ v_

where v+ and v_ are the stoichiometric coefficients of the anion and of thecation in the salt molecule. From Eqs (2.5.37) to (2.5.39) we then obtain

Z)+D_(v+ + v_)v_D+ + v+D_ 5 v ;

Diffusion of a single electrolyte ('salt') is thus characterized by aneffective diffusion coefficient

Replacement of the diffusion coefficients by the electrolytic mobilitiesaccording to Eq. (2.3.22) yields the Nernst-Hartley equation:

116

It could be expected for less dilute solutions that an expression for theeffective diffusion coefficient could be derived theoretically from Eq.(2.5.42) by using correction terms for U+ and U_ taken from theDebye-Huckel-Onsager theory of electrolyte conductivities. However, ithas been demonstrated experimentally that this approach does not lead to acorrect description of the dependence of the diffusion coefficient on theconcentration. The mobility of ions in the diffusion process varies far lesswith changes in the concentration than when charge is transported in anexternal electric field, and the effect of the concentration on the mobilitycan be either retarding, zero or accelerating, depending on the type of salt(increasing concentration always reduces ion mobility in an external electricfield). This difference between the two transport phenomena is a result ofthe fact that diffusion is connected with the movement of cations and anionsin the same direction, so that faster species are retarded by slower species,and vice versa, whereas during electric current flow, oppositely charged ionsmove in opposite directions and the two types of ions retard one another.The electrophoretic effect caused by the retardation of the movement of thecentral ion by the moving ionic atmosphere is thus of a different magnitudein diffusion. The time-of-relaxation effect is completely absent, as thesymmetry of the ionic atmosphere is not disturbed during diffusion.

As mentioned, the gradient of the diffusion electric potential is suppressedin the case of diffusion of ions present in a low concentration in an excess ofindifferent electrolyte ('base electrolyte'). Under these conditions, the simpleform of Fick's law (2.3.18) holds for the diffusion of the given ion. The

Table 2.4 Diffusion coefficients D x 106 (cm2 • s l) determined by means of polar-ography or chronopotentiometry at various indifferent electrolyte concentrationsc (mol • dm"3) at 25°C. The composition of the indifferent electrolyte is indicated for

each ion. (According to J. Heyrovsky and J. Kuta)

c

0.010.11.03.0

c

0.010.11.03.0

Ag+

KNO3

15.8515.3215.46

KNO3

6.606.386.20

KNO3

18.216.5

Zn2+

KC1

6.766.737.237.69

Tl+

KC1

17.415.713.5

NaOH

6.545.134.18

NaCl

17.715.09.2

IO3

KC1

10.159.899.36

Pb

KNO3

8.768.288.02

-

NaCl

10.018.927.24

2+

KC1

8.998.679.208.14

Fe(CN)6

KC1

6.506.506.20

Cd2+

KNO3 KC1

8.156.90 7.156.81 7.90

7.90

•- Fe(CN)63-

KC1

7.847.627.637.36

117

Table 2.5 Diffusion coefficients (cm2 • s l x 10 5) of electrolytes in aqueous solu-tions at various concentrations c (mol • dm~3). (According to H. A. Robinson and

R. H. Stokes)

Electrolyte

LiClNaClKC1KC1KC1RbClCsClLiNO,NaNO3

KC1O4

KNO,AgNO3

MgCl2

CaCl2

SrCl2

BaCl2

Li2SO4

Na2SO4

Cs2SO4

MgSO4

ZnSO4

LaCl3

K4Fe(CN)6

Temperature

(°C)

2525202530252525252525252525252525252525252525

0

1.3661.6101.7631.9932.2302.0512.0041.3361.5681.8711.9281.7651.2491.3351.3341.3851.0411.2301.5690.8490.8461.2931.468

0.001

1.3451.5851.7391.964

——

2.013——

1.8451.899

—1.1871.2631.2691.3200.9901.1751.4890.7680.7481.175

—

0.002

1.3371.5761.7291.954

—2.0112.000

—1.5351.8411.884

—1.1691.2431.2481.2980.9741.1601.4540.7400.7331.145

—

c

0.003

1.3311.5701.7221.9452.1742.0071.9921.296

—1.8351.8791.7191.1581.2301.2361.2830.9651.1471.4370.7270.7241.1261.213

0.005

1.3231.5601.7081.9342.1611.9951.9781.2891.5161.8291.8661.708

—1.2131.2191.2650.9501.1231.4200.7100.7051.1051.184

0.007

1.3181.555

—1.9252.1521.9841.9691.2831.5131.8211.8571.698

—1.2011.209

——————

1.084—

0.010

1.3121.5451.6921.9172.1441.9731.9581.2761.5031.7901.846

——

1.188—————————

diffusion coefficient, of course, depends on the concentration of theindifferent electrolyte.

A similar situation occurs in tracer diffusion. This type of diffusion occursfor different abundances of an isotope in a component of the electrolyte atvarious sites in the solution, although the overall concentration of theelectrolyte is identical at all points. Since the labelled and the original ionshave the same diffusion coefficient, diffusion of the individual isotopesproceeds without formation of the diffusion potential gradient, so that thediffusion can again be described by the simple form of Fick's law.

Experimental methods for determining diffusion coefficients are describedin the following section. The diffusion coefficients of the individual ions atinfinite dilution can be calculated from the ionic conductivities by using Eqs(2.3.22), (2.4.2) and (2.4.3). The individual diffusion coefficients of the ionsin the presence of an excess of indifferent electrolyte are usually found byelectrochemical methods such as polarography or chronopotentiometry (seeSection 5.4). Examples of diffusion coefficients determined in this way arelisted in Table 2.4. Table 2.5 gives examples of the diffusion coefficients ofvarious salts in aqueous solutions in dependence on the concentration.

118

2.5.5 Methods of measurement of diffusion coefficientsBoth steady-state and non-steady-state methods are used in the deter-

mination of diffusion coefficients. In the steady-state methods (dc/dt = 0)the flux J and the concentration gradient dc/Sx are measured directly andthe diffusion coefficient D is calculated according to Fick's first law, oftenwithout assuming that D is constant. In non-steady-state methods, all theparameters of diffusion flux change with position and time and thusequations resulting from integration of Fick's second law must be employedwith various boundary conditions, depending on the experimental arrange-ment. In integration, when necessary, a suitable form of the function D(x, t)must be assumed. Usually the diffusion coefficient is assumed to be constantand thus the final integrated equations can be used only for smallconcentration ranges in which the change in the diffusion coefficient can beneglected.

A steady-state method. In the diaphragm method the diffusion process isrestricted to a porous diaphragm (a glass frit with a pore size of aboutlOjUm, separating compartments 1 and 2 of volumes V1 and V2 (cf. Eq.(2.5.18). The diaphragm is the actual diffusion space with an effectivecross-section of pores A and an effective length /. The compartments arefilled with solutions of concentrations c? and c2 at the start of theexperiment, and are thoroughly stirred during the experiment. After a shorttime the solutions penetrate into the pores of the diaphragm and thesolution diffuses from 1 to 2 if c\>c\. In the steady state there is noaccumulation of the solute in the pores of the diaphragm, i.e. the sameamount of solute which leaves the compartment 1 in a certain time intervalpasses into the compartment 2. The flux of the substance is constant throughthe entire thickness of the diaphragm but it decreases with time because theconcentration difference between compartments 1 and 2 decreases. Afterthe experiment has been running for a certain time, the contents of bothcompartments are analysed and the concentrations c[ and c2 are determined.From these data we may compute the average value of the diffusioncoefficients between the concentrations (c? + c[)/2 and (c2 + c2)/2.

The time dependence of the concentration in 1 and 2 is obtained bysolving the differential equations

Integration using the relationship VxcQ

x + V2c2 = Vxcx + V2c2 yields

where At is the time interval during which concentrations c} and c2 assumevalues c[ and c2, respectively.

Fig. 2.14 The scheme of the cylindrical lens method for diffusion coefficient measurement: (1) the source with thehorizontal slit; (2) the condenser supplying a bandle of parallel beams; (3) the cuvette with a refraction index gradientwhere the beams are deflected; (4) the objective lens focusing the parallel beams to a single point; (5) the optical

member with an oblique slit and a cylindrical lens; (6) the photosensitive material

120

Non-steady-state methods. The most frequently used methods belongingto this category are the optical methods. The experiment is usually arrangedso that the boundary and initial conditions are fulfilled for linear diffusion ina limited space (tube) with one end closed and impermeable for thediffusing substance (Ddc/dx = 0) and the other end in contact withconstantly stirred pure solution. After an initial delay the gradient of therefractive index dn/dx which is directly proportional to the concentrationgradient is measured at several points of the cuvette. A parallel beam oflight incident upon the cuvette at a right angle to the direction of theconcentration gradient will be deflected in a direction towards the region oflarger refractive index and the angle of deflection will be directly propor-tional to the concentration gradient. The resulting dependence of thisdeviation on the position in the cuvette (Fig. 2.14) is then analysed on thebasis of theoretical relationships.

References

Covington, A. K., and M. J. F. Rebello, Reference electrodes and liquid junctioneffect in ion-selective electrode potentiometry, Ion-Sel. Electrode Revs., 5, 93(1983)

Crank, J., The Mathematics of Diffusion, Clarendon Press, Oxford, 1964.Cussler, E. L., Multicomponent Diffusion, Elsevier, Amsterdam, 1976.Cussler, E. L., Diffusion: Mass Transfer in Fluid Systems, Cambridge University

Press, Cambridge, 1984.Delahay, P., New Instrumental Methods in Electrochemistry, Interscience, New

York, 1954.Geddes, A. L., and R. B. Pontius, Determination of diffusivity, in Technique of

Organic Chemistry (Ed. A. Weissberger), Vol. 1, Part 2, 3rd ed., Interscience,New York, 1960, p. 895.

Marchiano, S. L., and A. J. Arvia, Diffusion in absence of convection. Steady stateand nonsteady state, CTE, 6, 65 (1983).

Morf, W. E., The Principles of Ion-Selective Electrodes and of Membrane Transport,Akademiai Kiado, Budapest, and Elsevier, Amsterdam, 1981.

Newman, J., Electrochemical Systems, Prentice-Hall, Englewood Cliffs, 1973.Overbeek, J. Th. G., The Donnan equilibrium, in Progress in Biophysics and

Biophysical Chemistry, Vol. 6, Pergamon Press, London 1956.

2.6 The Mechanism of Ion Transport in Solutions, Solids, Melts andPolymers

Diffusion is one of the most important natural processes. The varying rateof diffusion yields an excellent characteristic of individual states of aggrega-tion. While diffusion coefficients in gases are of the order of 10"1 cm2 • s~\those in solutions are of the order of 10~5cm2-s~1 and, in solids, of

The mobility of a species w, is a parameter connecting diffusion andmigration processes. This section will be concerned primarily with thequalitative theory of mobility, which has characteristic properties for

121

transport in dilute electrolyte solutions in electrically neutral liquids, inmelts (ionic liquids) and in solid substances.

2.6.1 Transport in solution

The transport of dissolved species in a solvent occurs randomly throughmovement of the Brownian type. The particles of the dissolved substanceand of the solvent continuously collide and thus move stochastically withvarious velocities in various directions. The relationship between themobility of a particle, the observation time r and the mean shift (x2) isgiven by the Einstein-Smoluchowski equation (in three-dimensional case)

The value of the mean shift of an ion (r2) does not change markedly, evenwhen a not exceedingly large electric field (<105 V • m"1) is applied to thestudied system.

The dissolved species are located within the quasicrystalline solventstructure and vibrate around equilibrium positions. After several hundredvibrations they attain a sufficiently great energy to permit a jump to aneighbouring position in the quasicrystalline lattice if a sufficiently large freespace is available. Thus, the frequency of jumps depends on the energyrequired to release a species from its equilibrium position and on the energyrequired to form a 'hole' in a neighbouring position in the solvent. Forsimple ions, the frequency of these jumps in common solvents is about1010-10n s"1 at normal temperatures.

When an electric field is applied, jumps of the ions in the direction of thefield are somewhat preferred over those in other directions. This leads tomigration. It should be noted that the absolute effect of the field on theionic motion is small but constant. For example, an external field oflV-m" 1 in water leads to ionic motion with a velocity of the order of50 nm • s"1, while the instantaneous velocity of ions as a result of thermalmotion is of the order of 100 m • s"1.

The velocity of motion of the particles depends on their dimensions andshape, on the interaction (e.g. association) between the solvent moleculesand finally on the interaction between particles of the dissolved substanceand solvent molecules. Consider the simplest case, where the molecule ofthe dissolved substance is much larger than the solvent molecule, isspherical and the interaction between the solute molecules and the solvent isnegligible. Then the motion of the particles of the solute can be consideredas the motion of spherical particles with radius r, through a viscous mediumwith viscosity coefficient r\. The velocity v is then described by the Stokeslaw.

^ (2.6.2)

122

where f, is the force acting on the particle. For diffusion, this velocitycorresponds to the material flux; thus similar considerations as thoseemployed for convection mass flux (Eq. (2.3.24)) lead to the relationship

= c,-vdif (2.6.3)

where vdif is the diffusion velocity.In Eq. (2.6.2), f, is the force acting on a single particle, whereas in Eq.

(2.3.20) grad \i is the force acting on 1 mole of particles. Thus,

Substitution of Eq. (2.6.4) into (2.6.2) and then into (2.6.3) yields

_ gradji,-(J/)dif - - 7 TT (2.6.5)

6nr\rNleading to the Stokes-Einstein equation for the diffusion coefficient,

kTA = 7 (2.6.6)

and to the relationship for the electrical mobility

^ (2.6.7)

The values derived in this way for the diffusion coefficients exhibitsurprising agreement with the experimental values, even for small ions,better than a coincidence in the order of magnitude. More detailedtheoretical analysis indicates that the formation of 'holes' required forparticle jump is analogous to the formation of holes necessary for viscousflow of a liquid. Consequently, the activation energy for diffusion is similarto that for viscous flow.

Wirtz et al. obtained better agreement for the diffusion coefficients thanthat yielded by Eq. (2.6.6), by considering the change in viscosity in thevicinity of the diffusing particle. The semiempirical equation has the form

kTD=—— (2.6.8)

where £ is the microviscosity factor

£ = 0.16 + 0 . 4 - (2.6.9)

with rL as the effective radius of a solvent molecule, obtained from themolar volume.

Using the statistical mechanical theory, Longuet-Higgins and Popleobtained a correlation function for the velocities in a liquid consisting of

123

rigid spheres. McGall, Douglas and Anderson modified this correlationfunction to yield an expression for the diffusion coefficient

(2.6.10)

where § is the fraction of the total volume occupied by solvent molecules.It follows from Eqs. (2.6.6), (2.6.8) and (2.6.10) that the presence of the

solvent has two effects on the ionic mobility: the effect of changing viscosityand that of changing the ionic radius as a result of various degrees ofsolvation of the diffusing particles. If the effective ionic radius does notchange in a number of solutions with various viscosities and if ionassociation does not occur, then the Walden rule is valid for these solutions:

= constant (2.6.11)

or

Arj= constant (2.6.12)

As the radii of solvated ions show no striking difference, the diffusioncoefficients for individual ions in water vary in the range 5 x 10~10 to2x l (T 9 m 2 - s - 1 (5x l ( r 6 cm 2 s - 1 to 2 x 10"5cm2s-1, cf. Table 2.4) andtheir electrolytic mobilities U{ in the range 2 x 10"8 to 10~7 m2 • s"1 • V"1.

The oxonium ion H3O+ in aqueous solutions has a very different diffusioncoefficient value, about five times higher than would be expected on thebasis of its dimensions. Similarly, the hydroxide ion also has a much largerdiffusion coefficient value than predicted. Analogously, the mobilities of theH3SO4

+ and HSO4~ ions in concentrated sulphuric acid are 50 and 100 timeshigher than those of the other ions. This effect is explained by the fact thatthe lyonium and lyate ions need not migrate through a protic solvent, but'move' through exchange of a proton between neighbouring solventmolecules (the so-called Grotthus mechanism of charge transport). It shouldbe recalled that Grotthus knew nothing about ions when he developed histheory of electrolytic decomposition (1809), but assumed that the transportof hydrogen and oxygen in solution during the electrolysis of water occursthrough alternate splitting of the water molecule, shift of the atoms ofhydrogen and oxygen produced and reformation of a water molecule.

According to contemporary concepts, a proton acts as a quantummechanical species during transfer between two neighbouring molecules(see Fig. 2.15) and the actual transfer occurs as a result of tunnellingthrough a potential energy barrier between the initial and final states of thesystem: H3O+ + H2O—»H2O + H3O+. However, tunnelling can occur onlywhen both species are favourably oriented. The activation energy for protontransfer depends on the rotation energy of a water molecule. This conceptof proton transfer between two water molecules is analogous to the transferof an electron between the oxidized and reduced forms of an ion, forexample Fe3+ + e ^ F e 2 + , or to the transfer of an electron between an

124

Fig. 2.15 Transport of protons inwater. Proton tunnelling is a fastprocess but water molecules mustfirst rotate to the position where the

transfer is possible

electrode and these species (see Section 5.3.1). The energy required for anorientation of the water molecule suitable for proton transfer plays a rolesimilar to the reorganization energy of the solvation shell involved in theseprocesses.

Eigen pointed out that the mobility of a proton in ice at 0°C is about 50times larger than in water. In ice, H2O molecules already occupy fixedpositions suitable for accepting a proton, so that the proton mobility isdirectly proportional to the rate of tunnelling.

2.6.2 Transport in solids

Diffusion and migration in solid crystalline electrolytes depend on thepresence of defects in the crystal lattice (Fig. 2.16). Frenkel defects originatefrom some ions leaving the regular lattice positions and coming to interstitialpositions. In this way empty sites (holes or vacancies) are formed, somewhatanalogous to the holes appearing in the band theory of electronic conduc-tors (see Section 2.4.1).

Ions are transferred in the lattice in three ways:

1. By a shift of an ion from one interstitial position to another

©0 © 0 © © ©© © ©© © 0 / © 0 y © © ©© O © 0 © ©0© ©1©© © X© 0 0 © © GrO© © ©0 © © 0© 0 ©

A B

Fig. 2.16 (A) Schottky and (B) Frenkeldefects in a crystal

125

2. By a shift of an ion from a stable position into an interstitial position andof another ion from an interstitial position into the hole formed

3. By a shift of an ion from a stable position into a neighbouring hole,forming a new vacancy

Current is conducted by the Frenkel mechanism, e.g. in silver halides,where the charge carrier is the silver ion (transport number tAg+ = 1).Impurities in crystals favour the Frenkel mechanism. Important materialswith high conductivity depending on the Frenkel mechanism include/J-alumina, a ceramic material with the composition Na2OllAl2O3 . Thestructure of j3-alumina (see Fig. 2.17) consists of a compact arrangement ofblocks of the y-phase of aluminium oxide of a thickness of four oxygenlayers and of a spinel structure. These blocks are separated by bridginglayers, containing only oxygen and sodium ions. Each bridging layercontains only one-fourth as many oxygen atoms as the compact layer inblocks. These are arranged so that the sodium ions are located in relativelysmall concentrations in tunnels between the oxygen ions, where they canmigrate relatively freely, leading to the unusually high conductivity ofj3-alumina. The conductivity mechanism depends on the Frenkel defects inthe bridging layer (interstitial oxygen ions close to interstitial aluminiumions in the compact block).

Other ceramic ion conductors similar to /3-alumina were introduced byGoodenough et al. in 1976; they belong to the so-called Nasicon family.Nasicon is a solid solution of the composition Na3Zr2Si2PO12, exhibiting aNa+ conductivity of 0.2S/cm at 300°C. Numerous Nasicon derivatives,

to tn i?

i ) • CW~CM~

O #O #0.

rOT) f )

Fig. 2.17 A scheme of /3-aluminastructure. The gaps between blocksof j8-phase A12O3 function as bridg-ing layers sparsely populated byoxygen (O) and sodium (•) ions.

(According to R. A. Huggins)

126

containing Sc, Cr, Fe, In, Yb, Ti, V . . . , etc. were also prepared andcharacterized as promising solid conductors for high-temperature batteries.The Nasicon structure is composed of two ZrO6 octahedra and three(P/Si)O4 tetrahedra in a rather loose packing with opened conductionchannels. Sodium cations are distributed inside these channels in twosymmetrically non-equivalent positions, Na(l) and Na(2), but only some ofthe available positions are occupied. The migration of Na+ ions throughthese channels is therefore rather easy and the observed ionic conductivitiesare higher by orders of magnitude in comparison with those of otherceramic conductors.

In some ionic crystals (primarily in halides of the alkali metals), there arevacancies in both the cationic and anionic positions (called Schottkydefects—see Fig. 2.16). During transport, the ions (mostly of one sort) areshifted from a stable position to a neighbouring hole. The Schottkymechanism characterizes transport in important solid electrolytes such asNernst mass (ZrO2 doped with Y2O3 or with CaO). Thus, in the presence of10 mol.% CaO, 5 per cent of the oxygen atoms in the lattice are replaced byvacancies. The presence of impurities also leads to the formation ofSchottky defects. Most substances contain Frenkel and Schottky defectssimultaneously, both influencing ion transport.

For Schottky defects in an ionic crystal with a cubic lattice, the diffusioncoefficient is given by the relationship, e.g. for a cation,

D = WPp (2.6.13)

where a is the distance between the closest cation and anion, fi is thefraction of vacancies in the total number of cations and p is the probabilityof the jump of a vacancy per second, given as

(2.6.14)

where v is the vibrational frequency of the cations and e] is the activationenergy for a jump.

Crystals with Frenkel or Schottky defects are reasonably ion-conductingonly at rather high temperatures. On the other hand, there exist severalcrystals (sometimes called 'soft framework crystals'), which show surpris-ingly high ionic conductivities even at the room or slightly elevatedtemperatures. This effect was revealed by G. Bruni in 1913; two well knownexamples are Agl and Cul. For instance, the ar-modification of Agl (stableabove 146°C, sometimes denoted also as ^-modification') exhibits at thistemperature an Ag+ conductivity (t+ = 1) comparable to that of a 0 . 1 Maqueous solution. (The solid-state Ag+ conductivity of ar-Agl at the meltingpoint is actually higher than that of the melt.) This unusual behaviour canhardly be explained by the above-discussed defect mechanism. It has beenanticipated that the conductivity of or-Agl and similar crystals is described

127

by a qualitatively different transport model, the so-called disorderedsub-lattice motion.

Every ionic crystal can formally be regarded as a mutually interconnectedcomposite of two distinct structures: cationic sublattice and anionic sub-lattice, which may or may not have identical symmetry. Silver iodideexhibits two structures thermodynamically stable below 146°C: sphalerite(below 137°C) and wurtzite (137-146°C), with a plane-centred I" sub-lattice. This changes into a body-centred one at 146°C, and it persists up tothe melting point of Agl (555°C). On the other hand, the Ag+ sub-lattice ismuch less stable; it collapses at the phase transition temperature (146°C)into a highly disordered, liquid-like system, in which the Ag+ ions are easilymobile over all the 42 theoretically available interstitial sites in the I~sub-lattice. This system shows an Ag+ conductivity of 1.31 S/cm at 146°C(the regular wurtzite modification of Agl has an ionic conductivity of about10~3 S/cm at this temperature).

Attempts have been made to lower the temperature of appearance of thesub-lattice motions. It was found that substitution in the I~ sub-lattice ofAgl, e.g. by WO|~, stabilizes this structure up to rather low temperatures:crystals of (AgI)1_A.(Ag2WO4)A. show, for x = 0.18, an Ag+ conductivity of0.065 S/cm at 20°C. Addition of cationic species, for instance in Ag2HgI4,Ag4RbI6, and Ag7[N(CH3)4]I8 has a similar effect.

Lanthanum fluoride (and fluorides of some other lanthanides) has anunusual type of defect (see Section 6.3.2), namely Schottky defects of themolecular hole type (whole LaF3 molecules are missing at certain sites).Charge carriers (F~) are formed as the result of interaction of LaF3 with thishole, leading to dissociation with formation of LaF2

+ and F~.Only the alkali metal ions are mobile in oxygen-containing glasses such

as the silicates or borates of the alkali metals. The relatively large free spacein the glass permits a jump from one oxygen atom to another withsimultaneous charge transfer between the oxygens forming bridges betweenthe silicon or boron atoms.

Similarly, as in the transport of ions in electrolyte solutions, random ionmotion predominates over ordered motion in the direction of the fieldduring the passage of electric current through solid substances.

2.6.3 Transport in melts

The mobility of ions in melts (ionic liquids) has not been clearlyelucidated. A very strong, constant electric field results in the ionic motionbeing affected primarily by short-range forces between ions. It would seemthat the ionic motion is affected most strongly either by fluctuations in theliquid density (on a molecular level) as a result of the thermal motion ofions or directly by the formation of cavities in the liquid. Both of thesepossibilities would allow ion transport in a melt.

128

2.6.4 Ion transport in polymers

Ion conducting polymers (polymer electrolytes) attract increasing atten-tion not only from the academic point of view, but also for their prospectiveapplications in batteries, sensors, electrolysers and other practical devices.Although the ion conducting polymers can formally be regarded as 'solidelectrolytes', the mechanism of ion transport is different from that in theabove-discussed inorganic crystals with lattice defects, and it resembles theion transport in liquid media. This follows from the fact that ions aretransported in a polymeric host material, which is essentially not as rigid asthe defect crystal of the classical inorganic solid electrolyte, i.e. the hostmotions or rearrangements virtually contribute to the ion transport as well.The ion conducting polymers therefore present a special class of electrolyteswith features intermediate between those of solid (defect crystals) and liquid(solutions, melts) electrolytes. The ion conducting polymers contain,besides the organic polymeric backbone, ions in more or less stronginteractions with the polymer, and in some cases also solvent molecules.

Ion solvating polymers. Polymers that are conducting even in the drystate, i.e. without attached solvent molecules, are termed ion solvating {orion-coordinating) polymers. These contain electronegative heteroatoms(prevailingly oxygen, sometimes also nitrogen, sulphur, or phosphorus),which interact with cations by donor-acceptor bonds similar to those insolvated ions in solutions. The complexes of ions with polymers are simplyformed by dissolving inorganic salts in suitable polymeric hosts. The mostimportant solvating polymeric hosts are two polyethers: poly(ethyleneoxide) (PEO) and poly(propylene oxide) (PPO):

—CH—CH2—O—

CH3

PEO PPO

Lithium perchlorate is dissolved in PEO owing to the coordination of Li+

cation by oxygen atoms in the polymeric chain. The complex thus formedhas the helical structure (Fig. 2.18), and exhibits ionic conductivities of uptol(T4S/cmat60oC.

The mechanism of ion transport in such systems is not fully elucidated,but it is presumably dependent on the degree of crystallinity of thepolymeric complex (which further depends on the temperature and the salttype). The ionic conductivity was initially attributed to cation hoppingbetween fixed coordination sites in the depicted helical tunnel, i.e. in thecrystalline part of the polymer.

A more detailed analysis revealed, however, that the cation hopping

cio;

129

CIO4

Fig. 2.18 Structure of a poly(ethylene)oxide-LidO4 complex

between vacant sites along crystalline regions in the polymer has onlymarginal significance for ion transport. The main arguments are: (1)completely amorphous polymers are also conducting, usually even betterthan the crystalline ones, and (ii) not only cations, but also anions are themobile species in ion-solvating polymers. (For instance, the Li+ transportnumber in the PEO-LiClO4 complex is in the region of only 0.2-0.35, anddrops, moreover, with increasing Li/PEO ratio.) The preconception of animmobile polymer host skeleton is also far from reality, especially foramorphous polymers.

The transition between crystalline and amorphous polymers is charac-terized by the so-called glass transition temperature, Tg. This importantquantity is defined as the temperature above which the polymer chains haveacquired sufficient thermal energy for rotational or torsional oscillations tooccur about the majority of bonds in the chain. Below Tg, the polymer chainhas a more or less fixed conformation. On heating through the temperatureTg, there is an abrupt change of the coefficient of thermal expansion (or),compressibility, specific heat, diffusion coefficient, solubility of gases,refractive index, and many other properties including the chemicalreactivity.

In polymer electrolytes (even prevailingly crystalline), most of ions aretransported via the mobile amorphous regions. The ion conduction shouldtherefore be related to viscoelastic properties of the polymeric host anddescribed by models analogous to that for ion transport in liquids. Theseinclude either the 'free volume model' or the 'configurational entropymodel'. The former is based on the assumption that thermal fluctuations ofthe polymer skeleton open occasionally 'free volumes' into which the ionic(or other) species can migrate. For classical liquid electrolytes, the freevolume per molecule, uf, is defined as:

v{=v- v0(2.6.15)

where v is the average volume per molecule in the liquid and v0 is the vander Waals' volume of the molecule. For polymers, the free volume, uf, can

130

also be expressed as:

vf=vg[k + a(T-TJ] (2.6.16)

where vg is the volume per molecule at the glass transition temperature, Tg,a is the thermal expansion coefficient of the polymer, and A: is anon-dimensional constant. The ion transport in the polymer is conditionedby the formation of sufficiently large voids in the polymeric skeleton. Theprobability of this process, p, decreases exponentially with the total-to-freevolume ratio, v/v{:

p=exp(-v/vf) (2.6.17)

Equations (2.6.16) and (2.6.17) can be rearranged to the form:

p = e x p [ - B / ( r - r 0 ) ] (2.6.18)

where B and To are constants (To is usually 20-50°C lower than 7^).Using Eq. (2.6.18) the temperature dependence of various transport

properties of polymers, such as diffusion coefficient D, ionic conductivity aand fluidity (reciprocal viscosity) 1/rj are described, since all these quantitiesare proportional to p. Except for fluidity, the proportionality constant(pre-exponential factor) also depends, however, on temperature,

D~Tm; o~T-m'p; 1/rj-p (2.6.19)

Another quasi-thermodynamical model of ion transport in polymers isbased on the concept of minimum configurational entropy required forrearrangement of the polymer, giving practically identical a — T and D — Tdependences as the preceding model.

The common disadvantage of both the free volume and configurationentropy models is their quasi-thermodynamic approach. The ion transport isbetter described on a microscopic level in terms of ion size, charge, andinteractions with other ions and the host matrix. This makes a basis of thepercolation theory, which describes formally the ion conductor as a randommixture of conductive islands (concentration c) interconnected by anessentially non-conductive matrix. (The mentioned formalism is applicablenot only for ion conductors, but also for any insulator/conductor mixtures.)

The main conclusion of the percolation theory is that there exists a criticalconcentration of the conductive fraction (percolation threshold, c0), belowwhich the ion (charge) transport is very difficult because of a lack ofpathways between conductive islands. Above and near the threshold, theconductivity can be expressed as:

o=o0(c-c0)" (2.6.20)

where the exponent n is about 1.2-1.5, depending on the spatial dimen-sionality of the system under study.

Besides the poly ether-based polymer electrolytes, the nitrogen analogues(polyimines) were also extensively studied. Various other oxygen-containing

131

polymers were further used, e.g. polyesters, polymers containing oxygenheterocycles, crown ethers, etc. Most of the studied polymer electrolytes arereasonably conducting at elevated temperatures, typically 60-100°C.Recently, proton conductors based on PEO complexes with anhydrousacids or acid salts, such as H3PO4 or NH4HSO4 were prepared; they showconductivities in the range 10~5-10~4S/cm even at the room temperature.Other highly conducting ion-solvating polymers contain etheric solvatinggroups on an inorganic polymeric skeleton, either phosphazene orsiloxane. The former have been prepared by chlorine substitution inpoly(dichlorophosphazene), (—N=PC12—)„. Some phosphazene deriva-tives show very low glass transition temperatures, and are thus regardedas promising ion-solvating polymers. The best known representative ispoly[bis(methoxyethoxyethoxide)] phosphazene:

O—(CH2)2—O—(CH2)2—O—CH3

( _ N = P _ ) n

O—(CH2)2—O—(CH2)2—O—CH3

also abbreviated as MEEP. This polymer forms complexes withLi+CF3SOJ, exhibiting a room-temperature ionic conductivity of up tol(T4S/cm.

The ion solvating polymers have found application mainly in powersources (all-solid lithium batteries, see Fig. 2.19), where polymer el-ectrolytes offer various advantages over liquid electrolyte solutions.

Polymer gels and ionomers. Another class of polymer electrolytes arethose in which the ion transport is conditioned by the presence of alow-molecular-weight solvent in the polymer. The most simple case is theso-called gel polymer electrolyte, in which the intrinsically insulatingpolymer (agar, poly(vinylchloride), poly(vinylidene fluoride), etc.) isswollen with an aqueous or aprotic liquid electrolyte solution. The polymerhost acts here only as a passive support of the liquid electrolyte solution, i.e.ions are transported essentially in a liquid medium. Swelling of the polymerby the solvent is described by the volume fraction of the pure polymer in thegel (Vp). The diffusion coefficient of ions in the gel (Dp) is related to that inthe pure solvent (Do) according to the equation:

Dp = A) exp [-kVp/(l - Vp)] (2.6.21)

where k is a constant.Another example of ion conducting polymer/ion/solvent systems are

polyelectrolytes based on ion-exchange polymers, also called ionomers. Theionic conductivity of ion-exchange polymers is usually very low in the drystate, but increases abruptly by orders of magnitude upon addition of a

132

small amount of a liquid solvent (several molecules per one immobilizedionic group). The most important representatives of ionomers are sulphon-ated fluoropolymers, such as Nafion:

where m is about 13.This material was first synthesized in the middle 1960s by E.I. Du Pont de

Nemours and Co., and was soon recognized as an outstanding ion conductorfor laboratory as well as for industrial electrochemistry. The perfluorinatedpolymeric backbone is responsible for the good chemical and thermalstability of the polymer. Nafion membrane swollen with an electrolytesolution shows high cation conductivity, whereas the transport of anions isalmost entirely suppressed. This so-called permselectivity (cf. Section 6.2.1)is a characteristic advantage of Nafion in comparison with classicalion-exchange polymers, in which the selective ion transport is usually not sopronounced.

Thanks to its high stability and permselectivity, Nafion has been used as aNa+ conductor in membrane electrolysis of brine in the chlor-alkaliindustry. This application was introduced in the early 1980s and is by far themost important use of ionomer membranes.

Materials similar to Nafion containing immobilized —COO" or —NR3+

groups on a perfluorinated skeleton were also synthesized. These areavailable in the form of solid membranes or solutions in organic solvents;the former can readily be used as solid electrolytes in the so called solidpolymer electrolyte (SPE) cells, the latter are suitable for preparingion-exchange polymeric films on electrodes simply by evaporating thepolymer solution in a suitable solvent.

The first Nafion-SPE cells were constructed in the later 1960s, i.e. longbefore the industrial membrane electrolysis of brine. These cells have atypical thin-layer arrangement with swollen membrane functioning as anelectrolyte without additional separate liquid phases. A Nafion membraneacts here as a separator of hydrogen and oxygen catalytic electrodes andsimultaneously as a solid electrolyte conducting H+ ions formed at theanode. The interelectrode spacing is very narrow, typically about 0.3 mm;minimizing ohmic losses and permitting relatively high current densities tobe attained.

Inhibition of anion transport in Nafion was attributed to the in-homogeneous structure of the ion exchange sites in the polymer network(Gierke cluster network model). It was found that Nafion contains (even in

133

seal

PEO.LiX

\ cathodeFig. 2.19 Scheme of an all-solid lithium battery with PEO based

electrolyte

the dry state) discrete hydrophilic regions, separated by narrow hydropho-bic channels. The hydrophilic regions are formed by a specific aggregationof the ionogenic side chains on the perfluorinated backbone, whereas thechannels are composed of hydrophobic fluorocarbon chains (see thepolymer formula above). The individual macromolecular chains are thusinterconnected by the embedded clusters into a more complicated supramo-lecular network (Fig. 2.20). This, moreover, explains why Nafion andrelated materials are practically insoluble in water, in spite of the absence ofregular interchain bonds as in classical cross-linked ion-exchange polymers(e.g. polystyrene sulphonate).

By swelling with aqueous electrolyte, cations (and, to lesser extent, alsoanions) penetrate together with water into the hydrophilic regions and formspherical electrolyte clusters with micellar morphology. The inner surface ofclusters and channels is composed of a double layer of the immobilized—SO^ groups and the equivalent number of counterions, M+. Anions inthe interior of the clusters are shielded from the —SO^ groups byhydrated cations and water molecules. On the other hand, anions are thus

-SO 3 M double layer electrolyte cluster

Fig. 2.20 The Gierke model of a cluster networkin Nafion. Dimensions are expressed in nm. Theshaded area is the double layer region, containingthe immobilized —SO3 groups with correspondingnumber of counterions M+. Anions are expelled

from this region electrostatically

134

'trapped' inside the clusters owing to the high electrostatic barrier in thehydrophobic channel where the shielding effect is suppressed. This makes itclear why the channels are preferentially conductive for cations, i.e. thepolymer membrane (despite the presence of free anions) is not anion-conducting.

Diffusion of cations in a Nafion membrane can formally be treated as inother polymers swollen with an electrolyte solution (Eq. (2.6.21). Particu-larly illustrative here is the percolation theory, since the conductive sites caneasily be identified with the electrolyte clusters, dispersed in the non-conductive environment of hydrophobic fluorocarbon chains (cf. Eq.(2.6.20)). The experimental diffusion coefficients of cations in a Nafionmembrane are typically 2-4 orders of magnitude lower than in aqueoussolution.

References

Conway, B. E., Proton solvation and proton transfer in solution, MAE, 3, 43(1964).

Dekker, A. J., Solid State Physics, Prentice-Hall, Englewood Cliffs, 1957.Eisenberg, A. E., and H. L. Yeager (Eds), Perfluorinated Ionomer Membranes,

ACS Symposium Series 180, ACS, Washington, 1982.Glasstone, S., H. Eyring, and K. J. Laidler, Theory of Rate Processes, McGraw-

Hill, New York, 1942.Hertz, H. G., Self-diffusion in solids, Ber. Bunsenges., 75, 183 (1971).Hladik, J. (Ed.), Physics of Electrolytes, Academic Press, New York, a multivolume

series, published since 1972.Huggins, R. A., Ionically conducting solid state membranes, AE, 10, 323 (1977).Inman, D., and D. C. Lovering (Eds), Ionic Liquids, Plenum Press, New York,

1981.Kittel, C , see page 104.Laity, R. W., Electrochemistry of fused salts, /. Chem. Educ, 39, 67 (1962).Linford, R. G., Electrochemical Science and Technology of Polymers, Vol. 1 & 2,

Elsevier, London, 1987 and 1990.MacCallum, J. R., and C. A. Vincent (Eds), Polymer Electrolyte Reviews, Elsevier,

London, 1987.Vashista, P., J. N. Mundy, and G. K. Shenoy (Eds), Fast Ion Transport in Solids,

North-Holland, Amsterdam, 1979.Vincent, C. A., The motion of ions in solution under the influence of an electric

current, /. Chem. Educ, 53, 490 (1976).

2.7 Transport in a Flowing Liquid

2.7.1 Basic concepts

Processes involving transport of a substance and charge in a streamingelectrolyte are very important in electrochemistry, particularly in the studyof the kinetics of electrode processes and in technology. If no concentrationgradient is formed, transport is controlled by migration alone; convection

135

Fig. 2.21 Prandtl and Nernst layer formationat the boundary between a solid plate and

flowing liquid (c* = 0)

has no effect. The opposite situation where concentration gradients areformed during transport has interesting characteristics; if transport throughmigration can be neglected (e.g. at a sufficiently high concentration of anindifferent electrolyte), transport is controlled by diffusion in the streamingliquid, i.e. convective diffusion is involved.

To a major part this chapter will consider processes occurring in a steadystate, i.e. where dcjdt = 0. It will be simultaneously assumed that the liquidis incompressible. A change in its velocity occurs close to the boundary witha solid phase or with another liquid. This is because the molecules of aliquid in direct contact with the solid phase have the same velocity as thesolid phase. In other words, their velocity relative to the solid phase is zero.It is therefore convenient to use a reference frame that is fixed with respectto the solid phase. Viscosity forces act on the layers of the liquid that arefurther apart from the phase boundary, forming a velocity distribution inthe liquid, depicted in Fig. 2.21 for liquid flowing along a solid plate. Atangent drawn from the origin to the curve of the dependence of thetangential velocity component, vx, on the distance y from the plateintercepts the straight line vx = Vo at a distance <50. The quantity 50 is termedthe thickness of the hydrodynamic or Prandtl layer. It should be pointed outthat <50 is a function of the distance from the leading edge of the plate, /. As/ increases, the quantity <50 also increases; the liquid also moves in adirection perpendicular to the plate, away from the plate with velocity vy.

If diffusion occurs towards the surface of the plate, the motion of theliquid at a distance from the surface inside the hydrodynamic layer issufficient to maintain the original concentration in the bulk of the solution,c°. A decrease of the concentration to a value c* at the phase boundary

136

(in Fig. 2.21 c* = 0) occurs only in a thin layer within the Prandtl layer, withthickness 6 < d0. The concentration c* is attained in the immediate vicinityof the interface and is determined by its properties (e.g. for transport to anelectrode, the electrode potential is decisive—see Section 5.3). The quantityS is termed the thickness of the Nernst layer. For a steady-state concentra-tion gradient in the immediate vicinity of the phase boundary we have

The steady-state distribution of the concentration of the diffusing substanceat the phase boundary is termed the Nernst layer.t

For the material flux (cf. Eq. 2.5.18) we have

Jy=^r (c° - c*) = K(C° - c*) (2.7.2)

where D is the diffusion coefficient and K = D/6 is the mass transfercoefficient (this term is employed in chemical and electrochemicalengineering).

2.7.2 The theory of convective diffusion

The material flux during convective diffusion in an indifferent electrolyte(grad 0 — 0) can be described by a relationship obtained by combination ofEqs (2.3.18) and (2.3.23):

J = Jdif + Jconv = -D grad c + cv (2.7.3)

where Jdif is the diffusion and Jconv the convection material flux.Equation (2.2.10) for ip = c and the incompressibility condition

divv = 0 (2.7.4)

yield

-^ = DV2c - v grad c (2.7.5)

or, for the steady state,DV2c - v grad c = 0 (2.7.6)

In Cartesian coordinates, this equation becomes

(d2c d2c d2c\ ( Be dc 3c\\dx dy dz I \ dx y dy dz)

f Nernst studied the theory of heterogeneous reactions around the year 1900 and assumed thatthe flow of a liquid along a phase boundary with a solid phase involves formation of animmobile liquid layer at this surface. A steady-state concentration gradient is then formedwithin this layer as a result of diffusion in the limited space (see Eq. 2.5.18). In fact, thediffusion process has the character of convective diffusion everywhere as a result of the velocitydistribution in the hydrodynamic layer.

137

The assumption of a steady state corresponds to laminar flow of theliquid. This is fulfilled only under certain conditions (limited velocity of theliquid, smooth phase boundary between the flowing liquid and the otherphase, etc.). Otherwise, turbulent flow occurs, where the local velocitydepends on time, pulsation of the system, etc. Mathematically the turbulentflow problem is a very difficult task and it is often doubtful whether, inspecific cases, it is possible to obtain any solution at all.

Liquid flow originates from the effect of various forces. In forcedconvection, these are mechanical forces acting, e.g. on a piston forcing theliquid into a vessel, or a stirrer transferring the impulse to a liquid in avessel, etc. In this manner, pressure gradients are formed in the solution,resulting in motion of the liquid. Natural {free) convection results whendensity changes are produced in the solution as a result of concentrationchanges produced by transport processes. It is the force of gravity thatcauses natural convection and produces hydrostatic pressure gradients insolution that are different from conditions in a liquid of constant densitythat is at rest.

The relationship between the motion of a viscous liquid and thementioned forces is given by the Navier-Stokes equation

— + (v grad)v = - - grad/? + vV2v + g^P Po^ (2.7.8)dt p p

where p is the variable liquid density, p0 is the density of the liquid at rest,v is its kinematic viscosity (v = r]/p), gis the acceleration of gravity and p isthe difference between the total pressure and the hydrostatic pressure. Theright-hand side of this equation describes the pressure gradient for forcedconvection, the effect of viscosity forces and the effect of gravitation duringnatural convection. For sufficiently intense forced convection, the effect ofnatural convection can be neglected. These conditions are assumed in thesubsequent discussion. As it is assumed that the liquid is incompressible, thecontinuity equation (2.7.4) is valid.

In forced convection, the velocity of the liquid must be characterized by asuitable characteristic value V{), e.g. the mean velocity of the liquid flowthrough a tube or the velocity of the edge of a disk rotating in the liquid,etc. For natural convection, this characteristic velocity can be set equal tozero. The dimension of the system in which liquid flow occurs has a certaincharacteristic value /, e.g. the length of a tube or the longitudinal dimensionof the plate along which the liquid flows or the radius of a disk rotating inthe liquid, etc. Solution of the differential equations (2.7.5), (2.7.7) and(2.7.8) should yield the value of the material flux at the phase boundary ofthe liquid with another phase, where the concentration equals c*.

If a rigorous solution to this problem is at all possible, it consists of twoparts:

(a) Solution of the hydrodynamic part of the problem in order to obtain a

138

relationship for the flow velocity as a function of the spatial coordinates(i.e. solution of Eq. (2.7.8) with suitable boundary conditions) and

(b) Solution of Eq. (2.7.6) or Eq. (2.7.7) on the basis of a knownrelationship for v, or for vx, vy and vz

This approach is possible only if density gradients are not formed in thesolution as a result of transport processes (e.g. in dilute solutions).Otherwise, both differential equations must be solved simultaneously—avery difficult task.

Two examples will now be given of solution of the convective diffusionproblem, transport to a rotating disk as a stationary case and transport to agrowing sphere as a transient case. Finally, an engineering approach will bementioned in which the solution is expressed as a function of dimensionlessquantities characterizing the properties of the system.

As a rotating disk is a very useful device for many types of electrochemi-cal research, convective diffusion to a rotating disk, treated theoretically byV. G. Levich, will be used here as an example of this type of transportprocess. Consider a disk in the xz plane, rotating around the y axis withradial velocity co (see Fig. 2.22). If the radius of the disk is sufficiently larger

Fig. 2.22 Distribution of streamlines of the liquidat the surface of a rotating disk. Above: a sideview; below: a view from the direction of the disk

axis. (According to V. G. Levich)

139

than the thickness of the hydrodynamic layer, then the change of flow at theedge of the disk can be neglected. Solution of the hydrodynamic part of theproblem, carried out by von Karman and Cochrane, indicates that thethickness of the hydrodynamic layer is not a function of the distance fromthe centre of the disk y; its value in centimetres is given by the relationship

1/2

(2.7.9)

The velocity of the liquid in the direction perpendicular to the surface of thedisk in centimetres per second is described by the equation

vy = -0.51co3/2y-y2y2 (2.7.10)

It will be assumed that the concentration gradient in the directionperpendicular to the surface of the disk is much greater than in the radialdirection. Then Eq. (2.7.7) reduces to the form

dc _ d2c

The boundary conditions are

y-><*>, dc/dy=Q, c = c°

For c in moles per cubic centimetre and D in square centimetres persecond, the solution to this problem has the form

and

J=-D(^) =0.62D2/3v-l/6<ol/2(c°-c*) (2.7.13)

The thickness of the Nernst layer (see Eq. 2.7.1) is

6 = 1.6lD1/3vl/6co-l/2

Convective diffusion to a growing sphere. In the polarographic method(see Section 5.5) a dropping mercury electrode is most often used.Transport to this electrode has the character of convective diffusion, which,however, does not proceed under steady-state conditions. Convectionresults from growth of the electrode, producing radial motion of thesolution towards the electrode surface. It will be assumed that the thicknessof the diffusion layer formed around the spherical surface is much smallerthan the radius of the sphere (the drop is approximated as an ideal sphericalsurface). The spherical surface can then be replaced by a planar surface

(p\ = 0.62D-1/3v-1/6a>1/2(c° - c*) (2.7.12)

140

Fig. 2.23 Movement of the solution towards an expanding sphere modelled by aprism with an expanding plane as a base. When the base expands from the originalarea A ^ Q D , (equal to Aotf3) to the area A2B2C2D2 (AQt^3) then the point in thesolution originally at a distance xx must move to the distance x2 in order to keep aconstant volume of the prism Vo (i.e. to preserve the condition of the incom-presssibility of the solution, see Eq. 2.7.4). This model fits the actual situation only

when the distance xx is small {xx AJ±)

increasing in size according to the same laws as those governing the growthof the surface of a sphere, while the solution phase is modelled by a prismwith an expanding plane as base. Let us first determine the velocity of apoint in the solution at the distance x from the surface (Fig. 2.23).

The original spherical surface increases as a result of an increase in itsvolume at a constant velocity u. The area of the surface of the sphere attime t is given by the relationship A = (4jt)V3(3ut)2/3 = A0t

2/3, where Ao isthe area at time t = \. From the incompressibility of the liquid it follows forvolume Aoxt2/3 that

(2.7.14)= constant

Differentiation of Eq. (2.7.14) with respect to time gives

dx 2xdxv = — = (2.7.15)

Substitution into Eq. (2.7.5) respecting only changes along the x coordinateyields the Ilkovic differential equation

dt(2.7.16)

with the initial condition c = c° and the boundary condition for x = 0,c = c*. Solution of this equation yields

where 7Hn is the material flux for linear diffusion (cf. Eq. 2.5.8). Theincrease in the rate of transport through convection is expressed by thecoefficient (7/3)1/2. This result is only an approximation based on simplifica-

141

tions in the derivation of Eq. (2.7.16). J. Koutecky gave a rigorous solutionfor diffusion to a growing sphere in the form of a power series. His solutionhas the form

/ = -v/-/lin(l + 1.04a-lDl/2t1/e) (2.7.18)

where a = (A0/4JZ)1/2 in centimetres is the radius of the sphere at timef = l s .

2.7.3 The mass transfer approach to convective diffusion

In hydrodynamic problems it is an advantage to transform all variablesinto dimensionless parameters by 'scaling' each variable, using constantscharacterizing the dimensions and other properties of a given system. Thus,in the differential equations describing the systems, these constants arecombined into dimensionless numbers or criteria. This formulation of theproblem is very convenient because a process may be characterized in amuch more general way by using the dimensionless criteria and, further-more, in more complicated cases it is possible to find the properties of asystem using a small model and then to apply the results to a large devicekeeping the criteria constant.

In consideration of the hydrodynamic problem alone, it is usuallyattempted to characterize the studied system by three quantities, thecharacteristic length / (e.g. the length of the plate in the direction of theflowing liquid or the radius of a rotating disk), the velocity of the flowingliquid outside the Prandtl layer Vo and the kinematic viscosity v.

The Cartesian coordinates of the system, x, y and z, are referred to thecharacteristic length / and the components of the velocity vx, vy and vz

within the Prandtl layer are referred to the characteristic velocity Vo,

-t=X, ^=Y, y = Z (2.7.19)

^=VU (2.7.20)

where u = x} y, z. When the first and third terms on the right-hand side ofEq. (2.7.8) are neglected, the Navier-Stokes equation is obtained inCartesian coordinates as three equations:

dX y dY z dZ Vol

The dimensionless number appearing in Eq. (2.7.21),

— = Re (2.7.22)

is called the Reynolds number.

142

Its value, among others, characterizes the flow of the liquid in the system.For not very large Reynolds numbers the flow is laminar, which means thatthe velocity is stationary and its change monotonic in a perpendiculardirection to the phase boundary between the phase boundary and the regionof maximum relative velocity of the liquid. For very large values of Re theflow becomes turbulent. The value of the Reynolds number characterizingthe start of turbulent flow depends strongly on the properties of the system.Thus, for example, in the case of a liquid flowing along a smooth plate thecritical value is Re —1.5 x 103. However, this limiting value may beconsiderably decreased by attaching various obstacles to the surface or byroughening the surface.

It holds approximately for the thickness of the Prandtl layer that

<50~/VRe (2.7.23)

In order for the flow along the boundary to come to a steady state, it isnecessary that R e » l . It follows from the form of Eq. (2.7.21) that thesolution for the flow rate can be expressed as a function of dimensionlessquantities:

Vu =fu(X, Y, Z, Re) (2.7.24)

The concentration of the transported substance c is converted to adimensionless form, C = c/(c° — c*). Then the equation for convectivediffusion at a steady state (2.7.7) can be converted by using Eqs (2.7.19) and(2.7.20) to the form

The quantity

-^- = Pe (2.7.26)

is the Peclet number. The ratio of the Peclet and Reynolds numbers iscalled the Schmidt number,

Pe v ,

The solution of Eq. (2.7.25) as the gradient of the dimensionlessconcentration C with respect to the dimensionless coordinate Y(perpendicular to the phase boundary) for Y = 0 is then given by therelationship

(!£) =<t>(X,Y,Re,Sc) (2.7.28)

143

Experimental measurements yield the mean value of the material flux JY,independent of the coordinates X and Y. If a further dimensionlessparameter, the Sherwood number, is introduced,

then Eqs (2.7.1) and (2.7.28) yield the criterial equation

Sh = F(Sc, Re) (2.7.30)

For the thickness of the Nernst layer we have (cf. Eq. 2.7.2)

6=i (2-7-31)In convective diffusion to a rotating disk, the characteristic velocity Vo isgiven by the product of the disk radius r, as a characteristic dimension of thesystem, and the radial velocity a>, so that the Reynolds number is given bythe equation

cor2

Re = (2.7.32)

It follows from Eq. (2.7.29) for / = r, and Eqs (2.7.13), (2.7.27) and (2.7.32)that

Sh = 0.62 Re1/2Sc1/3 (2.7.33)

When the relationship between the material flux and the parameters ofthe system can be calculated directly by solution of the appropriatedifferential equations, the criterion equation (2.7.30) has little significance.However, this is not possible in the great majority of practical systems, andthus the empirically determined criterial equation is of general validity forphysically similar systems. It can form a basis for designing largerequipment on the basis of experiments with model systems.

References

Agar, J. N., Diffusion and convection at electrodes, Faraday Soc. Discussions, 1, 26(1947).

Cussler, E. L., see page 20.Heyrovsky, J., and J. Kuta, Principles of Polarography, Academic Press, New York,

1966.Ibl, N., and O. Dossenbach, Convective mass transport, CTE, 6, 133 (1983).Levich, V. G., see page 20.Rousar, I., K. Micka, and A. Kimla, Electrochemical Engineering, Elsevier,

Amsterdam, Vol. 1, 1985, Vol. 2, 1986.

Chapter 3

Equilibria of Charge Transferin Heterogeneous Electrochemical

Systems

Again, the consideration of the electrical potential in the electrolyte, andespecially the consideration of the difference of potential in electrolyte andelectrode, involve the consideration of quantities of which we have no apparentmeans of physical measurement, while the difference of potential in pieces of metalof the same kind attached to the electrodes is exactly one of the things which we canand do measure.

J. W. Gibbs, 1899

3.1 Structure and Electrical Properties of Interfacial RegionsIn contrast to the previous chapters, dealing with the electrochemical

properties of a single phase, this and the next three chapters will beconcerned with electrochemical systems consisting of two or more phases incontact, at least one of which is an electronic or electrolytic conductor. Thesecond phase may be either another electrolytic conductor (this case will beconsidered the most extensively), or another electronic conductor, adielectric or a vacuum.

Two phases in contact are separated by a surface called a phase boundaryor interface (Fig. 3.1). The interface is unambiguously defined, for example,in the case of a metal-electrolyte solution interface. Surprisingly enough,even in the case of a liquid-liquid interface where the liquids are partiallymiscible the transition from one phase to the other is rather sharp.

The term 'interface' will be distinguished from a related term, theinterfacial region or interphase. This term denotes the region between thetwo phases where the properties vary markedly in contrast to those in thebulk of the phases. In the case of electrically conducting phases, chargedistribution occurs in this region.

The main criterion of the electrochemical properties of the interface iswhether electric charge can pass sufficiently rapidly across it in bothdirections, or whether it is impermeable to charge transfer and thus has the

144

145

phase phase ex

bulk

CCL

ion

o

ace

reg

terl

1 'su

r

i !

regi

face

bulk

interphase

Fig. 3.1 Structure of the interphase

properties of a capacitor without leakage. For the first type of interface, theelectrical potential difference between the two phases is a result of chargetransfer across the interface; these interfaces are termed non-polarizable. Inthis case, charge transfer involves the transfer of electrons from a metalconductor to an electron acceptor in the second phase, the emission ofelectrons into a liquid phase with formation of solvated electrons, thetransfer of ions from one phase into the other and similar processesoccurring in the opposite direction. In the second situation, chargedistribution occurs through charging of the two phases with opposite chargesfrom an external source, unequal adsorption of ions with opposite chargeson the two sides of the interface (including purely electrostatic attractionand repulsion of ions at the boundary between oppositely charged phases),adsorption and orientation of molecular dipoles and polarization of speciesin the inhomogeneous force field in the interfacial region. Such aninterface is termed polarizable. The term electrical (electrochemical) doublelayer is often used for the corresponding interface. An electrical doublelayer is, of course, present even at non-polarizable interfaces. In this case,however, it cannot be charged (i.e. charge separation cannot be attained) byintroduction of charge from an external source, but only through chargetransfer across the interface. The electrical double layer will be dealt with inChapter 4.

This chapter will include equilibria at non-polarizable interfaces for ametal or semiconductor phase-electrolyte system (a galvanic cell in thebroadest sense) and for two electrolytes (the solid electrolyte-electrolytesolution interface, or that between two immiscible electrolyte solutions).

3.1.1 Classification of electrical potentials at interfaces

The distribution equilibrium for uncharged species distributed betweentwo phases is determined by the equality of the chemical potentials of the

146

components in the two phases. The chemical potential of the ith componentin phase a is defined by the relationship (cf. Section 1.1.3)

(£Uwhere G' is the Gibbs energy, V is the internal energy of phase a and nt isthe amount of the given component (in moles). According to an alternativedefinition, the chemical potential is a measure of the work that must bedone under otherwise constant conditions for reversible transfer of onemole of uncharged species from the gaseous state with unit fugacity(assigned purely conventionally as zero on the chemical potential scale) intothe bulk of the given phase.

If one mole of charged particles is to be transferred from the givenreference state, further specified by the absence of an electric field, into thebulk of an electrically charged phase, then work must be expended not onlyto overcome chemical bonding forces but also to overcome electric forces.The work expended is given by jkif termed the electrochemical potential byGuggenheim and also defined by Eq. (3.1.1), where G' and U' are replacedby G and U; these are analogous to G' and U' and apply to an electricallycharged phase.

In general, the electrochemical potential cannot be separated intochemical and electrical components, as the chemical interaction of a specieswith its environment is also electric in nature. Nonetheless, the separationof the electrochemical potential is frequently made according to theequation

ft^. + zlty (3.1.2)

In this equation, the second term describes purely electrostatic work,connected with an infinitely slow transfer of charge zF from infinity in avacuum into the bulk of the second phase, i.e. to a point with electricpotential <j>. Here only electric charge is transferred, not a material specieswith which it might be connected. The ratio of this work to the transferredcharge is equal to the inner electrical potential <p of the given phase.

Equation (3.1.2) would imply separation of the effect of short-rangeforces (also including dipole interactions) and of the individual ionicatmospheres, related to \iiy from the long-range forces related to 0,identical with purely coulombic interaction between excess charges. It willbe seen later that such splitting, although arbitrary, is very useful.

The inner electrical potential (p may consist of two components. Firstly,the phase may possess some excess electrical charge supplied from outside.This charge produces an outer electrical potential ip. This is defined as thelimit of the ratio w/q for q—»0, where w is the work expended for theinfinitely slow transfer of charge q from an infinite distance to a point in thevacuum adjacent to the surface of the given phase and just outside the rangeof image forces. A particle transferred from this point further on in the

147

direction of the other phase must overcome the effect of electric forcesin the interphase. This component is called the surface electricalpotential %.

The component % is defined as the limit of the ratio w'/q for q—»0,where w' is the work expended in the transfer of charge q, from a point atwhich the outer electrical potential ip is defined, into the bulk of the phase.

The surface potential x consists of the contributions of ions present in theinterphase x(ion) and contributions from dipoles oriented in this region

(3.1.3)

(It should be recalled that the term 'surface potential' is used quite often inmembranology in rather a different sense, i.e. for the potential difference ina diffuse electric layer on the surface of a membrane, see page 443.) It holdsthat (/> = ip + x ( tms equation is the definition of the inner electricalpotential 0). Equation (3.1.2) can then be written in the form

fiL^M + zFip + zFx (3.1.4)

For uncharged species z = 0 and /2f- = \it. The first term in Eq. (3.1.2) isthus the chemical potential of a charged species in the phase considered.

If two phases oc and /?, with common species /, are brought into contact,then for uncharged species the tendency to pass from phase oc into phase j3 isexpressed by the difference ^(a) — jU,()3), and for charged species by thedifference &{&) — £/(|8). For equilibrium of uncharged species, ju^ar) =jM/(j8), while the condition for equilibrium of charged species is

fc(a) = |S,03) (3.1.5)

The result of the establishment of phase equilibrium is the formation ofdifferences of inner and outer potentials of both phases. These differencesare defined by the equations

*™-*™ (3.1.6)t

(3.1.7)

It should be noted that phase equilibrium can be attained for chargedspecies through far smaller transfer of particles from one phase into theother than for uncharged species, as the potential difference formed,0(/3) — 0(ar), decreases the original tendency for transfer between the twophases much faster than the changes in the original chemical potentials. Thedifference of the inner electrical potentials of the phases, A^0, is also calledthe Galvani potential difference and the analogous difference of the outerelectrical potentials, A^/T/;, is termed the Volta potential difference.

148

3.1.2 The Galvani potential difference

Equations (3.1.5) and (3.1.6) hold for any contact equilibrium of twophases with a common charged species. This species must be exchangedsufficiently rapidly between the two phases, so that the applicability of Eq.(3.1.5) is dependent on species i being transferred very quickly from phasea to phase jS and back again.

It should be noted that the Galvani potential difference consists of thecontribution of ions g(ion) and of the oriented dipoles g(dipole):

A£tf> = A^(ion) + Afe(dipole) (3.1.8)

In solid-state physics, the electrochemical potential of the electron fie(a)is mostly replaced by the equivalent energy of the Fermi level eF. While theelectrochemical potential is usually related to one mole of particles, theFermi energy is related to a single electron, so that

iWe=£FNA (3.1.9)

The lowest level of the conduction band for metals Vc and the highest levelof the valence band for semiconductors is very often used as a referencepoint for the energies of the Fermi levels; in this book, however, the energyof a free electron at rest in a vacuum will be used as the reference point forthe scale of the Fermi levels (cf. Fig. 3.2.).

The occupation of the energy levels of the conduction band in metals isdescribed by the Fermi function

exp kT(3.1.10)

where e is the energy of the given level. The Fermi energy corresponds to

£>0- £-0-

METAL

vacantlevels

N-TYPE

SEMICONDUCTOR

£-0-

P-TYPE

SEMICONDUCTOR

Fig. 3.2 Fermi level of isolated phases, metal, w-type semiconductor andp-type semiconductor, in a vacuum

149

reference vacuum level

a

Fig. 3.3 Fermi level of two metals in contact: (A) situation before thecontact when both phases have different Fermi levels, e£(ar) and e£(/3);(B) after coming into contact both the phases assume the same Fermi

level, eF

the just half-filled level. For semiconductors, the Fermi level lies in theforbidden band between the conductance and valence bands depending onthe presence of donor and acceptor levels.

At the contact of two electronic conductors (metals or semiconductors—see Fig. 3.3), equilibrium is attained when the Fermi levels (and thus theelectrochemical potentials of the electrons) are identical in both phases. Thechemical potentials of electrons in metals and semiconductors are constant,as the number of electrons is practically constant (the charge of the phase isthe result of a negligible excess of electrons or holes, which is incomparablysmaller than the total number of electrons present in the phase). The valuesof chemical potentials of electrons in various substances are of coursedifferent and thus the Galvani potential differences between various metalsand semiconductors in contact are non-zero, which follows from Eq. (3.1.6).According to Eq. (3.1.2) the electrochemical potential of an electron in

150

phase a isjUe(<*) = jUe(tf) — F(j)(a) (3.1.11)

Thus, for the Galvani potential difference between the metals a and /3 wehave

;pe(«)-^e(fl) ( 3_L l 2 )

If an electronic conductor is in contact with an electrolyte solutioncontaining the components of a simple redox system consisting of the ionicspecies Ox (charge numbers zOx) and Red (charge number zRed)

Ox + ze = Red (3.1.13)

where z = zOx — zRed (Fig. 3.4) then the relationship for the energy of theFermi level in the solution eF(J3) can be formulated as follows. The occupied

reference vacuum level

Ox

/7e(/3)

Red

reference vacuum level

a

Fig. 3.4 A metal in contact with a solution of an oxidation-reduction system.(A) Situation before the contact when the electrochemical potential ofelectrons in the electronic conductor (fi%a) = £F(<*)) has a different valuefrom the electrochemical potential of electrons in the oxidation-reductionsystem. (B) When the phases are in contact the electrochemical potential ofelectrons becomes identical in both a and ($ by charge transfer between them

151

energy levels are considered to be electron donor levels in the reducedform, Red, whose number is proportional to the activity of the reducedform, 0Red> while the unoccupied energy levels can be identified with theelectron acceptor levels in the oxidized form, Ox. Their number isproportional to the activity of the oxidized form, aOx. For the equilibrium ofreaction (3.1.13) we have

/ W £ ) + z£e(j3) = £Red(j3) (3.1.14)

Equation (3.1.2) and the definition of the chemical potential yield theequation for the electrochemical potential of species / with activity «/(/?) andcharge z, in phase jS in the form

fc(j8) = tffft) + RT\n at(P) + z,-/W) (3.1.15)In view of Eqs (3.1.13) and (3.1.14)

— i i - T \r~ / V

Combination of Eqs (3.1.2) and (3.1.16) yields a relationship for theGalvani potential difference between the metal and the solution,

= <t>{cc) — <t>(P)

(3.1.17)zF zF

We shall look more closely at this equation. On one hand, the standardchemical potentials of Ox and Red depend on their standard Gibbssolvation energies, AG^ox and AG° Red, and, on the other hand, on thestandard Gibbs energy of ionization of Red in the gas phase, AG|}on Red. Thisquantity is connected with the ionization potential of Red, /Rcd, which is,however, a sort of enthalpy so that it must be supplemented by the entropyterm, — TAS®on Ked. Thus, Eq. (3.1.17) is converted to the form

/Red - rASL,Red - AG(s),Red(j3) + zii%a)

zF

(3.1.18)

When the species common to the metal and the solution are metal ionsMz+, which are rapidly exchanged between the two phases, Eqs (3.1.5) and(3.1.15) readily yield the relationship for the Galvani potential differencebetween the metal and the solution

The equilibrium

152

exists in the metal phase, so that the standard chemical potential of themetal ions in the metal phase can be expressed as

I*M'+(<X) = J"M(*) + z+Pi%a) (3.1.20)

Substitution into Eq. (3.1.19) yields

V°M"(P) . RT+ — In aM*+(P) (3.1.21)

zrz+r +

If two electrolyte solutions in immiscible solvents are in contact and containa common ion Bz+ and the anions are not soluble in the opposite phase,then the relationship for the Galvani potential difference follows from Eq.(3.1.5):

z+F z+F laB^(a)i

When a solid electrolyte is in contact with an electrolyte solution, withcommon anion Az~(z_<0), then the Galvani potential difference betweenthe solid electrolyte and the solution is given by the equation

(3.L23)

Finally, consider a metal (ar)-solid electrolyte (jS)-electrolyte solution(y) system where the solid electrolyte and the electrolyte solution contain acommon anion A~ and the metal and the solid electrolyte contain acommon cation B+ (for simplicity, only univalent ions are considered). Thepotential difference between phase a (metal) and phase y (solution) is

RT.

The chemical potential of insoluble salt BA, fiBA, depends on its solubilityproduct according to the relationship (cf. Section 1.5.1)

&A = RT In PBA + MB+(y) + ^ - ( y ) (3.1.25)

so that Eq. (3.1.24) can be rearranged to give

& & RTIn PBA — I

RT RT

A"<t> = + — In PBA - — In aA-(y)

j y \naA(y) (3.1.26)

Two solid electrolytes in contact with a common ion show behaviouranalogous to two metals in contact.

153

As pointed out by J. W. Gibbs in 1875, thermodynamic procedurescannot be used to measure Galvani potential differences between chemicallydifferent phases (see page 144). Thus Eqs (3.1.12), (3.1.17), (3.1.18), (3.21)to (3.1.24) and (3.1.26) cannot be directly verified by measuring equilibriumquantities. However, their derivation was not pointless, because, as will beshown in Section 3.2, they have a form analogous to the relationships forequilibrium electrode potentials, consisting of several Galvani potentialdifferences.

Equilibrium measurements can be used only to determine the Galvanipotential differences between chemically identical phases, which is strictlypossible only for two identical electronic conductors. Then

A £ 0 = ^ (3.1.27)

as the 'chemical' parts of the electrochemical potential are identical.Potentiometric measurements (see Section 3.3.1) yield the differences A0for conditions expressed by Eq. (3.1.27). It will be demonstrated later thatuseful relationships have been deduced for potential differences betweenphases of different chemical composition on the basis of certain extrather-modynamic assumptions. This is especially true for measurements insolutions with liquid junctions (see Sections 2.5.3 and 3.1.4) and directmeasurement of the potential difference between two immiscible electrolytesolutions (Section 3.2.8).

3.1.3 The Volta potential difference

The outer electrical potential of a phase is the electrostatic potential givenby the excess charge of the phase. Thus, if a unit electric charge is broughtinfinitely slowly from infinity to the surface of the conductor to a distancethat is negligible compared with the dimensions of the conductor considered(for a conductor with dimensions of the order of centimetres, this distanceequals about 10~4cm), work is done that, by definition, equals the outerelectric potential ip.

In contrast to differences in the inner potentials (the Galvani potentialdifference), the difference in the outer potentials A^xp (the Volta potentialdifference or the contact potential) and the real potential at can bemeasured. The real potential at is defined as

(3.1.28)

The electrochemical potential can thus also be defined by the equation

) (3.1.29)

The real potential of electron times — 1, — ae(a)y is termed the electron workfunction O(ar). It should be realized that all these cases involve measure-ment of differences between two states. Since, however, one of these states

154

Table 3.1 Electron work function <£e(a) of some metals. (From J. Holzl, F. K.Schultze and H. Wagner)

Metal (with theMiller indices of thecrystal face)

Ag(100)Ag(llO)Ag(ll l)Cs

Cu (100)Cu(lll)Cu(110)Cu(112)K

Na

RbTa(110)Ta(l l l )Ta(l l l )

Oe(or) (eV)

4.64 ± 0.024.52 ±0.024.742.00

4.59 ±0.034.94 ± 0.054.48 ± 0.034.53 ± 0.032.26

2.46

2.054.8 + 0.6xl0~4T4.15 + 2 x l (T 4 T4.00 + 2.6 xlO~4T

Method

PhotoelectricPhotoelectricPhotoelectricContact potential

versus Ta (110)PhotoelectricPhotoelectricPhotoelectricPhotoelectricContact potential

versus Ta (110)Contact potential

versus Ta (110)PhotoelectricThermionic emissionThermionic emissionThermionic emission

is assigned a zero value by convention (as mentioned above, this value isassigned to a point infinitely far away from all the conductors), then we cansay that potentials or, and ip are measurable just as if they were absolutevalues. However, both the inner potential and the Galvani potentialdifference cannot be measured in the sense of 'absolute' values using purelythermodynamic procedures.

As a function of the surface potential %, the electron work function for agiven material depends on the state of the surface of that material(adsorption, the presence of surface compounds, etc.). For crystallinesubstances (see Table 3.1), various crystal faces have various electron workfunction values, which can be measured for single crystals. For poly crys-talline substances, the final value of the electron work function depends onthe contribution of the individual crystal faces to the entire area of thephase and the corresponding electron work functions; the final value of thework function, however, is strongly dependent on the experimental methodused for the measurement.

The electron work function is very important for the physics of the solidphase and for its application to electronics. In electrochemistry, it isespecially important for electrode processes involving adsorption of some ofthe species participating in the electrode reaction.

If two different, electrically uncharged phases are brought into contact,then the electrochemical potentials of their electrons become equal (i.e.

155

vacuum

— free surface

phase a phase (5

Fig. 3.5 The significance of the contact potential(Volta potential difference) A£V between two metals in

contact. (According to S. Trasatti and R. Parsons)

their Fermi levels are brought to the same level—see Fig. 3.2). An electricaldouble layer is formed on the contact surface and the outer potentials of thefree surfaces of both phases attain different values; i.e. a Volta potentialdifference (contact potential) is formed between them—see Fig. 3.5.

Measurement yields both the differences between the outer potentials andthe work functions (real potentials). If two phases oc an /? with a commonspecies (index i) come into contact, at equilibrium &(<*) = /*,•(/}), that is(Xi{a) — ai(P) = ZiFApij). These quantities are mostly measured using thevibrating condenser, thermoionic, calorimetric, and photoelectric methods.

The principle of the vibrating condenser method, originally proposed byThomson, is depicted in Fig. 3.6. Space A between the metal phases a, a'

p

r

i

ex1

G

A

a

P

Fig. 3.6 Thomson'svibrating condensermethod. A, space filledwith an inert gas at lowpressure; P, potentio-meter; G, null instrument

156

and /? is filled with an inert gas at very low pressure. Phases a and /? are incontact. Phases a and oc' are chemically identical. As they are surroundedby the same medium, x(a) = X(a') an<^ b°th the chemical potentialsand the electron work functions are identical for both these phases, that isjUe(ar) = jUe(ar'); ac(a) = ae(a'). The outer potentials of phases a' and (3 aredifferent, reflected in the fact that current flows through galvanometer G ona change in the distance between oc' and /3 or on ionization of atoms of thegas (e.g. by irradiation with a radioactive substance) in space A. Thiscurrent can be compensated by the voltage U derived from potentiometer P,for which

U = 4>(a) - 0("') = -F-\jie(*) - fie(a')) (3.1.30)

The condition at equilibrium is that il>(a') = ip(P) and fie(a) = jUe(/3) (thelatter equation describes the condition for contact equilibrium between theelectrons in phase a and jS). The measured compensating voltage \J,

U = F-\ac(a) - ajfi)) = y{a) - </>(£) (3.1.31)

thus yields both the difference of the electron work functions and thedifference of the outer electrical potentials of the two phases in contact, aand p.

The surface of the phases must be clean (one of the greatest experimentaldifficulties of this method); otherwise, the values of % would change alongwith the electron work functions.

The presence of impurities between phases oc and ]8 in contact implies theintroduction of a new phase (y). If current / passes and this new phasehas resistance R, then a further potential drop RI appears in the system.However, at equilibrium the presence of phase y has no effect on themeasured results, as it still holds that jWe(ar) = jUe(/J) = jUe(y). Similarly, thefinal equation is not affected by the fact that the leads and coils of thepotentiometer are not made of the same metal as phase a'. It can also bereadily seen that the validity of Eq. (3.1.31) is retained even in the presenceof a new phase y between phase a or phase a' and the potentiometer.

The difference in the outer potentials between the metal and theelectrolyte solution is measured similarly. In Fig. 3.6, phase /? will nowdesignate an electrolyte solution in contact with the metal phase a. Alsohere jUe(ar) = jUe(ar') and ip(a') = ^(P), and U is again expressed by Eq.(3.1.30); however, jUe(or):?t

iue(j8) as the communicating species betweenphases a and /J are not electrons but metal ions, Mz+. Thus,

£+(") = M 0 ) (3.1.32)

where the subscript + refers to the metal cations. However, in phase a,

tiM(a) = fi+{a) + z+fie(a) (3.1.33)

The subscript M refers to metal ions. An analogous procedure to that

157

employed above yields

F Z+F

= V>(")-V(0) (3-1.34)

Thus the compensating voltage U yields the difference between the outerelectrical potentials of the metal and the solution with which it is in contact.

The thermionic method is based on thermoemission of electrons in adiode. At elevated temperatures, metals emit electrons that are collectedat a gauze electrode placed opposite the metal surface and charged to a highpositive potential. C. W. Richardson has given the following relationship forthe saturation current /s:

(^) (3.1.35)

The temperature dependence of the current /s yields the electron workfunction —ac.

The calorimetric method is based on measuring the energy that must besupplied to the metal to maintain stationary emission of electrons. When ametal wire is heated to a high temperature, no electron emission startsunless the anode is connected to a high voltage source. Then, suddenemission of electrons produces a sharp temperature decrease as a result ofenergy consumption for release of the electrons. The difference between theamount of energy required to heat the wire to a certain temperature withand without electron emission depends on the work function for theelectrons of the given metal at that temperature.

The photoelectric method is based on the photoelectric effect. The kineticenergy of the electrons emitted during illumination of a metal with lighthaving a frequency v obeys the Einstein equation

\mv2 = hv- hv0 = hv- — (3.1.36)

where v0 is the limiting frequency below which no photoemission takesplace. The product of this frequency with the Planck's constant h gives thework function for a single electron, which can be determined when thewavelength of the light irradiating the metal is continuously changed.

3.1.4 The EMF of galvanic cells

A galvanic cell is a system that can perform electrical work when itsenergy is consumed at the expense of chemical or concentration changesthat occur inside the system. There are also systems analogous to galvaniccells based on conversion of other types of energy than chemical or osmoticinto electrical work. Photovoltaic electrochemical cells will be discussed in

158

Section 5.9.2. Thermogalvanic cells where electrical work is gained fromheat transfer will not be considered in this book and the reader is advised toinspect Agar's monograph quoted on page 169.

The electrode is considered to be a part of the galvanic cell that consistsof an electronic conductor and an electrolyte solution (or fused or solidelectrolyte), or of an electronic conductor in contact with a solid electrolytewhich is in turn in contact with an electrolyte solution. This definition differsfrom Faraday's original concept (who introduced the term electrode) wherethe electrode was simply the boundary between a metal and an electrolytesolution.

For example, consider a system in which metallic zinc is immersed in asolution of copper(II) ions. Copper in the solution is replaced by zinc whichis dissolved and metallic copper is deposited on the zinc. The entire changeof enthalpy in this process is converted to heat. If, however, this reaction iscarried out by immersing a zinc rod into a solution of zinc ions and a copperrod into a solution of copper ions and the solutions are brought into contact(e.g. across a porous diaphragm, to prevent mixing), then zinc will pass intothe solution of zinc ions and copper will be deposited from the solution ofcopper ions only when both metals are connected externally by a conductorso that there is a closed circuit. The cell can then carry out work in theexternal part of the circuit. In the first arrangement, reversible reaction isimpossible but it becomes possible in the second, provided that the otherconditions for reversibility are fulfilled.

In the reversible case, the total change of the Gibbs energy during thereaction occurring in the cell will be transformed into electrical work, whilein the irreversible case the electric work will be smaller and can be zero. Forreversible behaviour in a galvanic cell two requirements need to be fulfilled;there must be material and energetic reversibility. Most practical galvaniccells, however, show a certain degree of irreversibility.

Material reversibility can be demonstrated by balancing the EMF of thecell with an equal and opposite electrical potential difference from apotentiometer, and then causing the reaction in the cell (the cell reaction) toproceed, first in one direction and then in the reverse direction, by adjustingthe balancing electrical potential difference to be greater than and then lessthan the cell EMF.

Energetic reversibility is achieved when the same amount of electricalwork is supplied from the cell reaction proceeding in one direction as isgained from the reaction proceeding to the same degree in the oppositedirection.

Reversible operation of the cell requires that no other process occurs inthe cell than that connected with the current flow. An electrochemicalprocess that need not be always connected with the passage of current is thedissolution of a metal in an acid (e.g. zinc in sulphuric acid in the Volta cell)or the dissolution of a gas in an electrolyte solution (e.g. in a cell consistingof hydrogen and chlorine electrodes, hydrogen and chlorine are dissolved

159

and hydrogen chloride can be formed in solution even without current flow;this reaction, however, is strongly inhibited kinetically). Another processthat can occur in the absence of current passage is diffusion. In aconcentration cell (see Section 3.2.1), the concentrations of the two solutionscan be equalized, even in the absence of current passage, through diffusionalone. However, this part will consider cells for which both of the criteria ofthermodynamic reversibility are satisfied.

The irreversible processes described must not occur even on open circuit. Ina reversible cell, a definite equilibrium must be established and this may bedefined in terms of the intensive variables in a similar way to the descriptionof phase and chemical equilibria of electroneutral components.

Further, the passage of a finite amount of charge corresponds to a certaindegree of energetic irreversibility, as the electrical work is converted intoJoule heat because of the internal resistance of the cell and also in theexternal circuit; in addition, electrode processes involve a certain over-potential (see Chapter 5). Thus, the cell can work reversibly only onpassage of negligible current, i.e. when the difference between the balancingexternal voltage and the EMF of the cell approaches zero.

It should be noted that the reversibility of the galvanic cell has so farbeen considered from a purely thermodynamic point of view. 'Reversibleelectrode processes' are sometimes considered in electrochemistry in arather different sense, as will be described in Chapter 5.

A galvanic cell is usually depicted in terms of chemical symbols. Thephase boundary is designated by a vertical line. For aqueous solutions, thedissolved substances and their concentrations are indicated, and fornon-aqueous solutions, also the solvent. For example, the cell

Hg, Zn(12%) | ZnSO4(sat.), Hg2SO4(sat.) | Hg (3.1.37)

consists of a zinc amalgam electrode and a mercury electrode; the commonelectrolyte is a saturated solution of zinc and mercury sulphates.

As has already been mentioned, the EMF (the electromotive force) of acell is given by the potential difference between leads of identical metallicmaterial. In view of this, a galvanic cell is represented schematically ashaving identical metallic phases at either end.

The phases in the scheme are always numbered from the left to the right,but the cell can be depicted graphically in two ways. For example, there aretwo possibilities for the Daniell cell:

1 2 3 4 1'Zn | Zn2+ | Cu2+ | Cu | Zn W

1 2 3 4 1'Cu | Cu24" | Zn2+ | Zn | Cu K '

The choice between (a) and (b) is not specified, but does determine the signof the EMF of the cell. This quantity is always defined so that the electrical

160

potential of the last phase on the left (1) is subtracted from that of the lastphase on the right (1'). Thus, for the two given schemes of the Daniell cell,2sa = —Eb. For practical reasons it is preferable to place the positive electrode onthe right, as in scheme (a).

This section will consider only 'chemical galvanic cells' in which thechemical energy is converted to electrical energy.

The cell reaction is the sum of the electrochemical reactions taking placeat both electrodes. The cell reaction may be written in two ways which aredependent on the sequence of phases in the graphical scheme of the cell.The representation of the cell reaction should correspond to the flow of positivecharge through the cell (in a graphical scheme) from left to right:

Cu2+ + Zn -> Cu + Zn2+ (c)

Cu + Zn2+ -> Cu2+ + Zn (d)

Thus, Eq. (c) corresponds to scheme (a) and Eq. (d) to scheme (b). Theflow of the charge is due to the chemical forces described by the affinity of thecell reaction, and at the same time to the electrical forces measured by theelectrical potential difference. The affinity of the reaction is given by thealgebraic sum of the chemical potentials of all the substances participating inthe reaction multiplied by the corresponding stoichiometric coefficients(which are positive for reactants and negative for products); it is thus givenby the decrease in the Gibbs free energy of the system — AG. The sign ofthe affinity, however, is reversed when the reaction is written in theopposite direction. Thus, for reactions (c) and (d),

-AG c =+AG d (3.1.38)

If the reaction in the cell proceeds to unit extent, then the charge nFcorresponding to integral multiples of the Faraday constant is transportedthrough the cell from the left to the right in its graphical representation.Factor n follows from the stoichiometry of the cell reaction (for examplen = 2 for reaction c or d). The product nFE is the work expended when thecell reaction proceeds to a unit extent and at thermodynamic equilibriumand is equal to the affinity of this reaction. Thus,

^ and Eb = =j^ (3.1.39)

Considering that (-AGd) > 0 when (—AGC) < 0, then it is apparent that thesign of the EMF depends on the direction in which the scheme of the cell iswritten and, consequently, on the direction in which the reaction is written.The EMF is positive when the reaction and the scheme are formulated so asto correspond to a spontaneous reaction (the reaction would occurspontaneously in the direction indicated if the cell were short-circuited).Thus cell (a) has a positive EMF and cell (b) a negative one.

161

Consider a galvanic cell consisting of hydrogen and silver chlorideelectrodes:

Pt | H2 | HCl(c) | AgCl(s) | Ag | Pt (3.1.40)

In this cell, the following independent phases must be considered: platinum,silver, gaseous hydrogen, solid silver chloride electrolyte, and an aqueoussolution of hydrogen chloride. In order to be able to determine the EMF ofthe cell, the leads must be made of the same material and thus, to simplifymatters, a platinum lead must be connected to the silver electrode. It will beseen in the conclusion to this section that the electromotive force of a celldoes not depend on the material from which the leads are made, so that thewhole derivation could be carried out with different, e.g. copper, leads. Inaddition to Cl~ and H3O+ ions (further written as H+), the solution alsocontains Ag+ ions in a small concentration corresponding to a saturatedsolution of silver chloride in hydrochloric acid. Thus, the following schemeof the phases can be written (the parentheses enclose the species present inthe given phase):

1Pt(e)|

2|H 2 | solution(H

3 4-,Ag+)|AgCl(Ag+ ,C1")l

5Ag(Ag+

,e)

1'

1 Pt(e)(3.1.41)

The electrodes employed will be considered in greater detail in Section 3.2.Here it is sufficient that this cell can yield electrical energy, because the cellreaction

^H2(2) + AgCl(4)-> H+(3) + CT(3) + Ag(5) (3.1.42)

takes place spontaneously.Depending on the processes occurring during the cell reaction at the

individual electrodes, the cell reaction can be separated into two half-cellreactions formulated as reduction by electrons. For the cell reactiondescribed by Eq. (3.1.42), these reactions are

AgCl(4) + e = Ag(5) + Cl~(3) (3.1.43)

H+(3) + e = iH2(2) (3.1.44)

Subtraction of reaction (3.1.44) from reaction (3.1.43) yields reaction(3.1.42).

The EMF, E, is given by the difference in the inner potentials of phases Vand 1:

E = <t>(l')-<t>(l) (3.1.45)

If this EMF is compensated by an external electrical potential difference ofthe same magnitude and of opposite polarity, then no current flows and thesystem is at equilibrium. The phase equilibria of communicating species are

162

described as

fici,3 = Aci-,4; £Ag+,4 = MAg+,5; Ae,l = ]Ue,5 (3.1.46)

In the solid phases we have

^Ag,5 = Z^Ag+,2 + Ae,2

For the hydrogen half-cell reaction (Eq. 3.1.44),

2 ^ 3 + 2 ^ = ^ , 2 (3.1.48)

All the electrochemical potentials are expanded by using Eq. (3.1.1) intothe chemical potential and the inner electrical potential; the difference(3.1.45) is then found from Eqs (3.1.46) to (3.1.48), to be

FE = -jUH+,3 - Ma ,3 ~ MAg,5 + MAgCl,4 + 2j H2,2 (3.1.49)

The right-hand side of this equation expresses the affinity of reaction(3.1.42). In general, for a cell reaction proceeding according to a givenstoichiometric equation with stoichiometric factors v, and with transfer ofcharge equal to nF> that

-AG = - 2 v,ju, = nFE (3.1.50)

If the galvanic cell drives an external current, it supplies work to itssurroundings. For a reversible system, the difference between the externalelectrical potential difference and the EMF approaches zero and, in thelimiting case, these two potentials differ in sign only. Thus, the electricalwork supplied to the surroundings is given by the expression nFE = — AG,i.e. the decrease in the Gibbs free energy of the system.

It follows from Eq. (3.1.50) that the product nFE can be expressed by theGibbs-Helmholtz equation:

nFT( ^/dE\(ar) (3.1.51)\ ^ * ' p,composition

If the dependence of the EMF on the temperature is known, then thereaction enthalpy AH of the reaction proceeding outside the cell can befound. Cells with a positive EMF, E, and a negative temperature quotientdE/dT release heat in addition to electrical energy during reversibleprocesses

-TAS = -nFTl — \ (3.1.52)\ul / p composition

where A5 is the entropy change of the cell reaction.When producing electrical current on reversible operation, the cells with

both positive E and dE/dT absorb heat from their surroundings, as then-AH>nFE.

163

If the dependence of the EMF of a cell on temperature is known, thesame value of the reaction enthalpy AH can be determined as if the reactiontook place outside the cell.

3.1.5 The electrode potential

Equation (3.1.50) can be developed further. Consider once again cell(3.1.40), together with Eq. (3.1.49). We shall assume that hydrogen gas isunder standard pressure;t in addition, metallic silver, solid silver chlorideand gaseous hydrogen at standard pressure are selected as standards, that is

MH2 ,2O = 1) = /*H2, i"Ag,5 = i" Ag a n d MAgCl,4 = i^

Since also

MH+(W) = ju?I+(w) + RT In aH+(w),

Mcr(w) = fiMw) + RT l n «ci-(w), fC(w) + f*Mw)

and flH+(w)flci-(w) =

0±,HCI(W), we can write

FE = - ^ ( w ) ~ A*ci-(w) - 2RT ln a±tHCi(w)

2iKH (3.1.53)

(where the symbol w indicates that the substances are present in aqueoussolution). It is further assumed that the HC1 activity in cell (3.1.40) equalsunity. The quantity

H U " ) + & (w) + H° - H ° g - - V H 2

is the standard EMF of the cell. In the present case where the electrode onthe left in the graphical scheme is the standard hydrogen electrode, E° istermed the standard electrode potential. Standard electrode potentials aredenoted as E° with a subscript in which the initial substances in the half-cellreaction are given first, separated from the products by a slanted line; e.g.the standard potential of the silver-silver chloride electrode is designated as£Agci/Ag,ci-- For typographical reasons, however, it is prefable to includethe reactants in parentheses as follows:

£Agc,/Ag,cr - £°(AgCl + e = Ag + Cl") - £AgC,/Ag,cr (3.1.55)

It follows directly from the definition of the standard potential that, for thehydrogen electrode,

£H+/H2 = E\H+ + e = ^H2) - E°H+/H2 = 0 (3.1.56)

If the equilibrium constant of the cell reaction is denoted as K, then it

t It follows from Section 1.3.1 that the expression for the chemical potential must contain thepressure referred to the standard pressure; thus p = 1 corresponds to standard pressure,102 kPa. The activities in these expressions are also dimensionless and correspond to units ofmoles per cubic decimetre because of the choice of the standard state.

164

follows from Eq. (3.1.50) and from the equation for the standard reactionGibbs energy change AG° = -RT In K that

E°=(RT/F)\nK (3.1.57)

The relationships of the type (3.1.54) and (3.1.57) imply that the standardelectrode potentials can be derived directly from the thermodynamic data(and vice versa). The values of the standard chemical potentials areidentified with the values of the standard Gibbs energies of formation,tabulated, for example, by the US National Bureau of Standards. On theother hand, the experimental approach to the determination of standardelectrode potentials is based on the cells of the type (3.1.41) whose EMFsare extrapolated to zero ionic strength.

For example, the tabulated values for the reactants of the cellreaction (3.1.42) are AG°nAgci) = -109.68 kJ • moP1 and AG/

0(Ha,w) =

-131.14 kJ-mol"1, both at 298.15 K.t By definition, the standard Gibbsenergies of formation of H2 and Ag are equal to zero. Thus, the standardpotential of the silver-siiver chloride electrode is equal to

£Agci/Ag,cr = (-1.0968 x 105 + 1.3114 x 105): 9.648 x 104 = 0.2224 V.

It should be noted that Eq. (3.1.52) can be obtained from the expressionfor the Galvani potential difference formulated in Section 3.1.2. Thedesignation of the phases in the symbols for the individual Galvani potentialdifferences in cell (3.1.41) will be given in brackets. The overall EMF, E> isgiven by the expression

E= 0(1 ')- 0(1)= 0(1') - 0(5) + 0(5) - 0(4) + 0(4) - 0(3) + 0(3) - 0(1)= A '0 + A^0 + A^0 + A?0 (3.1.58)

The relationship for the individual Galvani potential differences can beobtained from Eqs (3.1.12), (3.1.24) and (3.1.17) so that Eq. (3.1.58) isconverted to the form (pH2=l)

/ig(Pt) - pg(Ag) - p V ( A g )h +

RT, M°H+(3)--i/x°H2(2) + /i°(Pt) RT- -=- In ac,-(3) —In aH+

r t t

& 2RT

i.e. a relationship identical with Eq. (3.1.53). The correct formulation of therelationships for the Galvani potential differences leads to the same results

tThe data have been recalculated from the previous standard pressure of 1.01325 x 105 to105 Pa and those determined for unit activity on the molal scale to unit activity on molar scale(cf. Appendix A).

165

as the procedure following from Eq. (3.1.39) or from Eqs (3.1.45) to(3.1.49).

It should be noted that all terms concerning the electrons in the metals aswell as those connected with the metals not directly participating in the cellreaction (Pt) have disappeared from the final Eq. (3.1.49). This result is ofgeneral significance, i.e. the EMFs of cell reactions involving oxidation-reduction processes do not depend on the nature of the metals where thosereactions take place. The situation is, of course, different in the case of ametal directly participating in the cell reaction (for example, silver in theabove case).

For cell reactions in general, both approaches yield the equation for theEMF:

RTE = E° ln£ (3.1.60)

nrwhere E° is the standard EMF (standard cell reaction potential), n is thecharge number and Q is the quotient of the activities of the reactants in thecell reaction raised to the appropriate stoichiometric coefficients; thisquotient has the same form as the equilibrium constant of the cell reaction.

For practical reasons it is often useful to separate the EMF of a galvaniccell into two terms and assign each of them to one of the electrodes. Thehalf-cell reactions provide a basis for unambiguous separation. The equationfor the overall EMF is converted to the difference between two expressions,in which the expression corresponding to the electrode on the left issubtracted from the expression corresponding to the electrode on the right.Thus, for example, it follows for Eq. (3.1.59), with inclusion of a termcontaining the relative hydrogen pressure, that

^ - ( w ) - RT In qcl-(w)

ln/7H2 + PH+(W) + RT In

(3.1.61)

The terms £Agci/Ag,ci- and EH+/H2 are designated as the electrodepotentials. These are related to the standard electrode potentials and to theactivities of the components of the system by the Nernst equations. By aconvention for the standard Gibbs energies of formation, those related tothe elements at standard conditions are equal to zero. According to afurther convention, cf. Eq. (3.1.56),

JI2,+(W) - AG}>(H+, w) = 0 (3.1.62)Then

) (3.1.63)

166

It follows from Eqs (3.1.53) and (3.1.55) that

RTEAgcuAg,ci- = Agci/Ag,ci- - — In «cr(w) (3.1.64)

rRT RT

Euvu2 = ~y{naH+~JP ln^H2 (3.1.65)

which are the Nernst equations. They will be discussed in detail in Section3.2. It should be recalled that, in contrast to standard electrode potentials,which are thermodynamic quantities, the electrode potentials must becalculated by using the extrathermodynamic expressions described inSections 1.3.1 to 1.3.4. The electrode potentials do not have the characterof Galvani potential differences, but are simply operational quantitiespermitting simple calculation or interpretation of the EMF of a galvanic celland are well suited to the application of electrochemistry in analyticalchemistry, technology and biology.

In the subsequent text the half-cell reactions will be used to characterizethe electrode potentials instead of the cell reactions of the type of Eq.(3.1.42) under the tacit assumption that such a half-cell reaction describesthe cell reaction in a cell with the standard hydrogen electrode on theleft-hand side.

The EMF of a cell is calculated from the electrode potentials (expressedfor both electrodes with respect to the same reference electrode) as thedifference of the potentials of these electrodes written on the right and leftin the scheme:

^cell = ^rhs — lhs (3 .1 .66)

In practice, it is very often necessary to determine the potential of a test(indicator) electrode connected in a cell with a well defined secondelectrode. This reference electrode is usually a suitable electrode of thesecond kind, as described in Section 3.2.2. The potentials of theseelectrodes are tabulated, so that Eq. (3.1.66) can be used to determine thepotential of the test electrode from the measured EMF. The standardhydrogen electrode is a hydrogen electrode saturated with gaseous hydrogenwith a partial pressure equal to the standard pressure and immersed in asolution with unit hydrogen ion activity. Its potential is set equal to zero byconvention. Because of the relative difficulty involved in preparing thiselectrode and various other complications (see Section 3.2.1), it is not usedas a reference electrode in practice.

The term 'electrode potential' is often used in a broader sense, e.g. forthe potential of an ideally polarized electrode (Chapter 4) or for potentialsin non-equilibrium systems (Chapter 5).

Similar to electrode potentials, standard electrode potentials have so farbeen referred to the standard hydrogen electrode (SHE). These data arethus designated by 'vs. SHE' after the symbol V, that is £Agci/Ag,ci- =

167

0.2224 V vs. SHE. Sometimes electrode potentials are referred to otherreference electrodes, such as the saturated calomel electrode (SCE), etc.

So far, a cell containing a single electrolyte solution has been considered(a galvanic cell without transport). When the two electrodes of the cell areimmersed into different electrolyte solutions in the same solvent, separatedby a liquid junction (see Section 2.5.3), this system is termed a galvanic cellwith transport. The relationship for the EMF of this type of a cell is basedon a balance of the Galvani potential differences. This approach yields aresult similar to that obtained in the calculation of the EMF of a cell withouttransport, plus the liquid junction potential value A0L. Thus Eq. (3.1.66)assumes the form

£cell = £rhs - £.hs + A0L (3.1.67)

However, in contrast to the EMF of a galvanic cell, the resultantexpressions contain the activities of the individual ions, which must becalculated by using the extrathermodynamic approach described in Section1.3.

Concentration cells are a useful example demonstrating the differencebetween galvanic cells with and without transfer. These cells consist ofchemically identical electrodes, each in a solution with a different activity ofpotential-determining ions, and are discussed on page 171.

It is very often necessary to characterize the redox properties of a givensystem with unknown activity coefficients in a state far from standardconditions. For this purpose, formal {conditional) potentials are introduced,defined in terms of concentrations. Definitions are not given unambiguouslyin the literature; the following would seem most suitable. The formal(conditional) potential is the potential assumed by an electrode immersed ina solution with unit concentrations of all the species appearing in the Nernstequation; its value depends on the overall composition of the solution. Ifthe solution also contains additional species that do not appear in the Nernstequation (indifferent electrolyte, buffer components, etc.), their concentra-tions must be precisely specified in the formal potential data. The formalpotential, denoted as EOf, is best characterized by an expression inparentheses, giving both the half-cell reaction and the composition of themedium, for example £°'(Zn2+ 4- 2e = Zn, 10"3M H2SO4).

Coming back to equations for Galvani potential differences, (3.1.11) to(3.1.26), we find that equation (3.1.67) can be written in the form

£cci = A^220 - Agj'0 + A0L (3.1.68)

where A™20 is the Galvani potential difference between the right-hand sideelectrode and the solution S2 in which it is immersed and A™'0 is theanalogous quantity for the left-hand side electrode. When the left-hand sideelectrode is the standard electrode and A0L is kept constant (see page 114),then Ecell can be identified with a formal electrode potential given by the

168

approximate relationship

+ constant (3.1.69)

Electrode potentials are relative values because they are defined as theEMF of cells containing a reference electrode. A number of authors haveattempted to define and measure absolute electrode potentials with respectto a universal reference system that does not contain a further metal-electrolyte interface. It has been demonstrated by J. E. B. Randies, A. N.Frumkin and B. B. Damaskin, and by S. Trasatti that a suitable referencesystem is an electron in a vacuum or in an inert gas at a suitable distancefrom the surface of the electrolyte (i.e. under similar conditions as those formeasuring the contact potential of the metal-electrolyte system). In thisway a reference system is obtained that is identical with that employed insolid-state physics for measuring the electronic energy of the bulk of aphase.

The system

M ; | S | M | M r (3.1.70)

where M is the studied electrode metal, S is the electrolyte solution andMr = M is the reference electrode metal, has the EMF

E = [<KMr) - <KM)] + [<KM) - <HS)] + [<KS) - <HMr)] (3.1.71)

Examination of Eq. (3.1.12) yields

0(Mr) - 0(M) = — — (3.1.72)

so that substitution into Eq. (3.1.69) yields

*-<Zf!\ (3.1.73,

Thus the EMF has been separated into two terms, each containing aquantity related to a single electrode. If the surface potential of theelectrolyte #(S) is added to each of the two expressions in brackets in Eq.(3.1.73), then the expression for the EMF contains the difference in theabsolute electrode potentials; for the absolute electrode potential of metalM we have

£M(abs) = A£V - ^ ^ + Z(S) (3.1.74)t

This quantity corresponds to the minimal work required to transfer anelectron from the bulk of metal M through the electrolyte solution into avacuum. Equation (3.1.74) can be readily modified to yield (cf. the

169

definition of the electron work function on page 153)

£M(abs) = —4r-^ + A£V (3.1.75)F

Both quantities, the electron work function Oe(M) and the contactpotential of the metal-electrolyte system A^T/J, are measurable quantities.

Unfortunately, as shown by Trasatti, because of water adsorption boththese values cannot escape a certain ambiguity. The absolute electrodepotential of the standard hydrogen electrode is most often reported as4.44 ± 0.02 V. Recent studies showed, however, that values between 4.2and 4.8 V can also be considered.

References

Agar, J. H., Thermogalvanic cells. AE, 3, 31 (1964).Holzl, J., F. K. Schultze, and H. Wagner, Solid Surface Physics, Springer-Verlag,

Berlin, 1979.Kittel, C , Introduction to Solid State Physics, John Wiley & Sons, New York, 1966.Manual of symbols and terminology for physico-chemical quantities and units.

Appendix III. Electrochemical nomenclature. Pure Appl. Chem., 37, 499 (1974).Parsons, R., Equilibrium properties of electrified interfaces, MAE, 1, 103 (1954).Reiss, H., The Fermi level and the redox potential, /. Phys. Chem. 89, 3783 (1985).Trasatti, S., The work function in electrochemistry, AE, 10, 213 (1977).Trasatti, S., The electrode potential, CTE, 1, 45 (1980).Trasatti, S., Structuring of the solvent at metal/solution interfaces and components

of electrode potential,/. Electroanal. Chem., 150, 1 (1983).Trasatti, S., and R. Parsons, Interphases in systems of conducting phases, Pure

Appl. Chem., 55, 1251 (1983).Trasatti, S., Structure of the metal/electrolyte solution interface: New data for

theory, Electrochim. Ada, 36, 1659 (1991).

3.2 Reversible Electrodes

Although from the thermodynamic point of view one can speak onlyabout the reversibility of a process (cf. Section 3.1.4), in electrochemistrythe term reversible electrode has come to stay. By this term we understandan electrode at which the equilibrium of a given reversible process isestablished with a rate satisfying the requirements of a given application. Ifequilibrium is established slowly between the metal and the solution, or isnot established at all in the given time period, the electrode will in practicenot attain a defined potential and cannot be used to measure individualthermodynamic quantities such as the reaction affinity, ion activity insolution, etc. A special case that is encountered most often is that ofelectrodes exhibiting a mixed potential, where the measured potentialdepends on the kinetics of several electrode reactions (see Section 5.8.4).

As demonstrated in Section 3.1, electrode equilibrium is a distributionequilibrium of charged species (including the electron) between the metal

170

and the solution, so that both phases must have at least one charged speciesin common. The chemical potentials of this species in the two phases thendetermine the magnitude of the potential difference between the metal andthe solution. Thus, the concept of a reversible electrode does not includethe limiting case where the concentration of the species determining theelectrode potential is zero in one of the phases (see Chapter 4).

We will now consider the principal electrodes that may be consideredreversible, and some of their combinations in galvanic cells. Reversibleelectrodes may be divided into three groups:

1. Electrodes of the first kind. These include cationic electrodes (metal,amalgam and, of the gas electrodes, the hydrogen electrode), at whichequilibrium is established between atoms or molecules of the substanceand the corresponding cations in solution (see Eqs 3.1.21 and 3.1.65),and anionic electrodes, at which equilibrium is established betweenmolecules and anions.

2. Electrodes of the second kind. These electrodes consist of three phases.The metal is covered by a layer of its sparingly soluble salt, usually withthe character of a solid electrolyte, and is immersed in a solutioncontaining the anions of this salt. The solution contains a soluble salt ofthis anion. Because of the two interfaces, equilibrium is establishedbetween the metal atoms and the anions in solution through two partialequilibria: between the metal and its cation in the sparingly soluble saltand between the anion in the solid phase of the sparingly soluble salt andthe anion in solution (see Eqs (3.1.24), (3.1.26) and (3.1.64)).

3. Oxidation-reduction electrodes. An inert metal (usually Pt, Au, or Hg)is immersed in a solution of two soluble oxidation forms of a substance.Equilibrium is established through electrons, whose concentration insolution is only hypothetical and whose electrochemical potential insolution is expressed in terms of the appropriate combination of theelectrochemical potentials of the reduced and oxidized forms, which thencorrespond to a given energy level of the electrons in solution (cf. page151). This type of electrode differs from electrodes of the first kind onlyin that both oxidation states can be present in variable concentrations,while, in electrodes of the first kind, one of the oxidation states is theelectrode material (cf. Eqs 3.1.19 and 3.1.21).

Ion-selective electrodes are based on membrane systems and will be dealtwith in Section 6.3.

3.2.1 Electrodes of the first kind

Electrodes of the first kind can be divided into anionic and cationic. Thesystem of a gas electrode includes a gas interacting with a suitable metal orsemiconductor surface in the cell reaction. However, gas electrodes can also

171

be considered as oxidation-reduction electrodes, as one of the forms can bea gas dissolved, perhaps in small concentration, in the solution. In amalgamelectrodes, the metal is present as an amalgam. The electrode potential thenalso depends on the gas pressure or on the activity of the metal in theamalgam.

The potential of a cationic electrode of the first kind corresponding to thehalf-cell reaction

M2+ + z+e = M

is described by the Nernst equation

RTE = E° + lna+ (3.2.1)

z+F

where z+ is the number of electrons required for reduction of one metal ion,i.e. its charge number (e.g. zinc in a solution of zinc(II) ions, copper in asolution of cupric ions, silver in a solution of silver(I) ions, etc.). It canreadily be demonstrated by the procedure described in Section 3.1 (page163) that the standard potential of a cationic electrode is given by theexpression

z+F(3.2.2)

The standard potentials of some electrodes of the first kind are listed inTable 3.2.

The combination of two electrodes of the same material differing only inthe solution concentration yields a concentration cell, for example

Ag | AgNO3(c1) | AgNO3(c2) | Ag (3.2.3)

Combination of Eqs (2.5.33), (3.1.67) and (3.2.1) and rearrangement (theconcentrations in Eq. (2.5.33) are replaced by the activities) yields the EMFfor a concentration cell with transport in the form (for z+ = z_ = 1):

^ RTt a+1 RT a+l RT a_xE = — ln^-f+-rln^-f_--ln--1 (3.2.4)

F 0+2 F 0+2 F 0_ 2

When eliminating the liquid junction potential by one of the methodsdescribed in Section 2.5.3, we obtain a concentration cell without transport.The value of its EMF is given simply by the difference between the twoelectrode potentials. More exactly than by the described elimination of theliquid junction potential, a concentration cell without transport can beobtained by using amalgam electrodes or electrodes of the second kind.

In amalgam electrodes, the metal is dissolved in mercury, so that not onlythe concentration of metal cations in the solution but also the concentrationof metal in the amalgam is variable. The potential of an amalgam electrode

172

Table 3.2 Standard potentials of electrodes of the firstkind at 25°C. (From the CRC Handbook of Chemistry andPhysics, recalculated to the standard pressure of 102 kPa)

Electrode

Li+/LiRb+/RbCs+/CsK+/KBa2+/BaSr2+/SrCa2+/CaNa+/NaMg2+/MgBe27BeA13+/A1Zn2+/ZnFe2+/FeCd2+/CdIn3+/InT1+/T1Sn2+/SnPb2+/PbCu2+/CuHg2+/HuAg+/Ag

E°(V)

-3.0403-2.98-2.92-2.931-2.912-2.89-2.868-2.71-2.372-1.847-1.662-0.7620-0.447-0.4032-0.3384-0.336-0.1377-0.1264

0.34170.8510.7994

Electrode

OH/O2(g)cr/ci2(g)F~/F2(g)

£°(V)

0.4011.357932.866

is given as

£ = E° + ^ l n ^ (3.2.5)zF a

where a is the activity of the metal in the amalgam and a+ is the activity ofits ions in the solution. Combination of two amalgam electrodes with thesame activity of cations in solution and different activities of the metal in theamalgam yields a different type of concentration cell, for example

K,Hg(fll) | KCl(m) | K,Hg(«2)with (3.2.6)

RFy ax£ = — l n -

F a2

If ax >a2, then short circuiting of this cell results in potassium dissolution atthe left-hand electrode and incorporation into the amalgam at the right-hand electrode. Amalgam electrodes can be used as reversible electrodes,even for metals as reactive as the alkali metals, especially in somenon-aqueous solvents.

Theoretically, the most important cationic electrode of the first kind is thehydrogen electrode. This is a gas electrode at which equilibrium is

173

established between hydrogen and oxonium ions in solution through aplatinum electrode coated with platinum black. This coating is obtained byelectrolytic deposition of platinum from an acidic solution of hexachloro-platinic(IV) acid and is saturated with hydrogen at a specific (usuallyatmospheric) pressure, pHr The platinum black has a double function.Firstly, its high specific surface area ensures the presence of a sufficientamount of hydrogen atoms adsorbed on the electrode; secondly, itfacilitates rapid establishment of the equilibrium between the oxonium ionsand elemental hydrogen (cf. Section 5.7.1). The hydrogen must be carefullypurified to remove traces of sulphur(II), arsenic and mercury compoundsthat would poison the electrode (cf. Section 5.7.2).

The potential of the standard hydrogen electrode, i.e. the potential valueat / ? H 2 = 1 and aH3O+ = l is set by convention as zero (cf. Eqs 3.1.62 and3.1.65). It can be seen from Eq. (3.1.65) that, when the activity of hydrogenions increases tenfold, the potential of the hydrogen electrode increases by2.303RT/F = 0.0591 V at 25°C. If the hydrogen pressure increases tenfold,then the potential decreases by 2.303RT/2F = 0.0295 V. Variation of thehydrogen pressure in the range of ±5 per cent corresponds to a potentialchange from +0.63 to —0.66mV; these differences are negligible for somemeasurements.

The hydrogen electrode is one of the electrodes suitable for pHmeasurements. If a hydrogen electrode saturated at unit pressure iscombined with a reference electrode, then the EMF of this cell (where thereference electrode is written on the left-hand side in the scheme of the cell)is given by the equation

E = £H+/H2 ~ ^ref = -1—= lOg «H+ ~ ^ref (3.2.7)t

and thus

(E + Eref)FPU~~ 2.303*7 ( 3 1 8 )

It would appear from Eq. (3.2.8) that the pH, i.e. the activity of a singletype of ion, can be measured exactly. This is not, in reality, true; even if theliquid junction potential is eliminated the value of Eref must be known. Thisvalue is always determined by assuming that the activity coefficients dependonly on the overall ionic strength and not on the ionic species. Thus themean activities and mean activity coefficients of the electrolyte must beemployed. The use of this assumption in the determination of the value ofEref will, of course, also affect the pH value found from Eq. (3.2.8). Thus,the potentiometric determination of the pH is more difficult than wouldappear at first glance and will be considered in the special Section 3.3.2.

The hydrogen electrode can be used to measure pH values over the wholepH region. However, it is not applicable to reducing or oxidizing media or

174

Fig. 3.7 A simple immersionhydrogen electrode

in the presence of substances that act as catalytic poisons for platinumblack. Poisoning of the electrode (p. 173) is indicated by fluctuation of theelectrode potential which differs from the value predicted by the Nernstequation.

Hydrogen electrodes can be constructed in variously shaped cells accord-ing to the purpose. An example is given in Fig. 3.7. The measurement isdescribed in detail in practical manuals. However, for practical measure-ments, the glass electrode (Section 6.3) is employed.

The potential of an anionic electrode of the first kind is given by therelationship

—Z-F \na. (3.2.9)

where z_ is the charge number of the corresponding anion (z_ < 0). Theequation also often contains a term including the effect of the gas pressureon a gas anionic electrode of the first kind.

The chlorine electrode contains gaseous chlorine in equilibrium withatomic chlorine adsorbed on the platinum black and a solution of chlorideions. Its potential is given by the equation

RT—rlt

RT—t (3.2.10)

The chlorine electrode behaves reversibly, i.e. electrode equilibrium is

175

Table 3.3 Standard electrode potentials in eutectic meltsreferred to a chlorine reference electrode. (According to

R. W. Laity)

System

Mn2+/MnCd2+/CdT1+/T1Co2+/CoPb2+/PbCu+/CuAg+/Au

-E

KCl-NaCl, 450°C

2.1351.535

1.2771.3521.1450.905

;°(V)

KCl-LiCl, 450°C

2.0651.5321.5861.2071.3171.0670.853

established rapidly; however, side reactions also occur to a small degree: theattack on the platinum electrode by chlorine and the reaction of chlorinewith water (C12 + H2O*±HOC1 4- H+ + Cl"), which can be suppressed insufficiently acidic medium.

In chloride melts, the chlorine electrode (with graphite instead ofplatinum) is used as a reference electrode (see Table 3.3).

Anionic electrodes of the first kind are rarely used in practice; other,more important, sorts of electrode exhibiting a reversible response to anionsare the electrodes of the second kind.

3.2.2 Electrodes of the second kind

The expression for the potential of electrodes of the second kind on thehydrogen scale can be derived from the affinity of the reaction occurring in acell with a standard hydrogen electrode. For example, for the silver chlorideelectrode with the half-cell reaction

AgCl(s) + e = Ag + C r (3.2.11)

^Agci/Ag = ^Agci/Ag — — l n 0 c r (3.2.12)

The same expression is obtained if electrodes of the second kind areconsidered as electrodes of the first kind, where the activity of the metalcations depends on the solubility product of the given insoluble salt (cf. Eq.3.1.26):

RT= EAg+/Ag + ~TT In ^Agci (3.2.13)

RT

It can be seen from this equation that the solubility product of silverchloride can be calculated from the known standard potentials of the silver

176

and silver chloride electrodes. Besides the silver chloride electrode thefollowing electrodes belong to this group:

The calomel electrode:

KCl(m) | Hg2Cl2(s) | HgRT

EHg2ci2/Hg = ^Hg2ci2/Hg - — In acr (3.2.14)

The mercurous sulphate electrode:

K2SO4(m) | Hg2SO4(s) | HgRT

^Hg2so4/Hg = ^Hg2so4/Hg ~ ~ In «so42- (3.2.15)

The mercuric oxide electrode:

KOH(m) | HgO(s) | HgRT

^Hgo/Hg = ^Hgo/Hg ~ In flOH- (3.2.16)

The latter does not yield a completely reproducible potential; equilibrium isestablished after about two days. Nonetheless, it is often used in research ontechnically important galvanic cells working in alkaline media.

The described electrodes, and especially the silver chloride, calomel andmercurous sulphate electrodes are used as reference electrodes combinedwith a suitable indicator electrode. The calomel electrode is used mostfrequently, as it has a constant, well-reproducible potential. It is employedin variously shaped vessels and with various KCl concentrations. Mostly aconcentration of KCl of 0.1 mol • dm"3, 1 mol • dm"3 or a saturated solutionis used (in the latter case, a salt bridge need not be employed); sometimes3.5 mol • dm"3 KCl is also employed. The potentials of these calomelelectrodes at 25°C are as follows (according to B. E. Conway):

0.1 M KCl | Hg2Cl2(s) | Hg 0.3337 V1 M KCl | Hg2Cl2(s) | Hg 0.2812 V

KCl(sat.) | Hg2Cl2(s) | Hg 0.2422 V

Table 3.4 Standard potentials of electrodesof the second kind. (From the CRC Hand-book of Chemistry and Physics, recalculated

to the standard pressure 102 kPa)

Electrode E° (V)

HgO, OH/HgAgCl, Cr/AgHg2Cl2, CP/HgHg2SO4, SO4

2 /HgPbO2, PbSO4, SO4

2~/Pb

0.09750.222160.267910.61231.6912

177

A B C DFig. 3.8 Various types of reference electrode vessels: (A) laboratory-type calomel electrode; (B), (C) portable calomel electrodes, (D) aAg/AgCl micropipette with a KC1 electrolyte immobilized with agar in

the tip of the capillary

The silver chloride electrode is preferable for precise measurements. Thestandard potentials of some electrodes of the second kind are listed in Table3.4. Various electrode constructions are shown in Fig. 3.8.

In addition to their use as reference electrodes in routine potentiometricmeasurements, electrodes of the second kind with a saturated KC1 (or, insome cases, with sodium chloride or, preferentially, formate) solution aselectrolyte have important applications as potential probes. If an electriccurrent passes through the electrolyte solution or the two electrolytesolutions are separated by an electrochemical membrane (see Section 6.1),then it becomes important to determine the electrical potential differencebetween two points in the solution (e.g. between the solution on both sidesof the membrane). Two silver chloride or saturated calomel electrodes areplaced in the test system so that the tips of the liquid bridges He at therequired points in the system. The value of the electrical potential differencebetween the two points is equal to that between the two probes. Similarpotential probes on a microscale are used in electrophysiology (the tips ofthe salt bridges are usually several micrometres in size). They are termedmicropipettes (Fig. 3.8D.)

3.2.3 Oxidation-reduction electrodesOxidation-reduction electrodes, abbreviated to redox electrodes, consist

of an inert metal (Pt, Au, Hg) immersed in a solution containing two forms

178

of a single substance in different oxidation states. The metal merely acts as amedium for the transfer of electrons between the two forms (cf. Section3.1.2). The redox electrode potential can be derived in an analogousmanner as for the previous types of electrodes (cf. Eq. (3.1.17)). For thecell reaction

Ox + ze = Red (3.2.13)

where z is the number of electrons necessary for reduction of one particle ofOx we obtain the Nernst equation

aRcd

(3.2.17)

This equation has been verified experimentally by Peters and is sometimesnamed after him.

For example, for the system Fe 3 + -Fe 2 + the Nernst equation is

£FC3+/Fe2+ = £FC3+/Fc2+ + — In — (3.2.18)

F aFe2+

In the half-cell reaction

Ox + mH + + ze = Red + nX (3.2.19)the change in the oxidation state is only displayed in the particles Ox andRed which may also function as acids or bases or as coordinationcompounds X denoting the ligand. Under these circumstances the Nernstequation has the form

^ox/Red = £ox,Red + (RT/zF) In - ^ (3.2.20)

The quinhydrone electrode (Section 3.2.5) is an example for such a morecomplicated redox reaction.

The standard potential is a measure of the reducing or oxidizing ability ofa substance. If E{{>E\> then system 1 is a stronger oxidant than system 2(similarly for metals, metal 1 would be more noble than metal 2), i.e. in anexperiment in the bulk where originally «Rcd,i=«ox,i and flRed,2 = «ox,2equilibrium would be established for which aRed,i > #ox,i and tfRed,2 < #ox,2-

The formal potential of a reduction-oxidation electrode is defined as theequilibrium potential at the unit concentration ratio of the oxidized andreduced forms of the given redox system (the actual concentrations of thesetwo forms should not be too low). If, in addition to the concentrations ofthe reduced and oxidized forms, the Nernst equation also contains theconcentration of some other species, then this concentration must equalunity. This is mostly the concentration of hydrogen ions. If the concentra-tion of some species appearing in the Nernst equation is not equal to unity,then it must be precisely specified and the term apparent formal potential isthen employed to designate the potential of this electrode.

179

E°-0.05 E°Electrode potential, Y

E*0.05

Fig. 3.9 Graphical representation of Eq. (3.2.21) with n indicated at eachcurve

Equation (3.2.17) can be expressed in terms of the degree of oxidation ocy

if the activities are set equal to the concentrations:

a^Ox/Red • - Ox/Red ' zF I-a

(3.2.21)

This equation is depicted graphically in Fig. 3.9. Differentiation of thepotential E with respect to a yields the following relationship at the point* = £:

\da/a=

ART

m- •

(3.2.22)

The slope of the tangent to the curve at the inflection point where a = \ isthus inversely proportional to the number of electrons n. The E-oc curvesare similar to the titration curves of weak acids or bases (pH-ar). Forneutralization curves, the slope dpH/dar characterizes the buffering capacityof the solution; for redox potential curves, the differential dE/dacharacterizes the redox capacity of the system. If a = ^ for a buffer, thenchanges in pH produced by changes in a are the smallest possible. If a = \in a redox system, then the potential changes produced by changes in a arealso minimal (the system is 'well poised').

It can be seen from Eq. (3.2.21) that, if the solution contains one of theforms, oxidized or reduced, in zero concentration, then the potential wouldattain infinite absolute value. These extreme values change to finite valuesin the presence of a trace of the other form. This always occurs as a result of

180

Table 3.5 Standard oxidation-reduction potentials.(From the CRC Handbook of Chemistry and Physics)

Half-cell reaction

-0.407-0.255

O2 + H2O + 2e <± HO2~ + OH" -0.076TiOH3+ + H+ + e *± Ti3+ + H2O -0.055

0.1510.1530.358

Ferricinium + + e<^ ferrocene 0.400I2 + 2e*±2r 0.5353Fe

3+ + e«±Fe2+ 0.771HO2~ + H2O + 2e *± 3OH~ 0.878Fe(phenanthroline)3

3+ + e *±Fe(phenanthroline)3

2+ 1.147Tl3+ + 2e<P*Tl+ 1.2152Mn3+ + e^Mn 2 + 1.5413

1.610

charge transfer from the electrode double layer to the component present insolution.

The standard potentials of some redox systems are listed in Table 3.5.

3.2.4 The additivity of electrode potentials, disproportionation

Electrode potentials are determined by the affinities of the electrodereactions. As the affinities are changes in thermodynamic functions of state,they are additive. The affinity of a given reaction can be obtained by linearcombination of the affinities for a sequence of reactions proceeding from thesame initial to the same final state as the direct reaction. Thus, the principleof linear combination must also be valid for electrode potentials. Theelectrode oxidation of metal Me to a higher oxidation state z+>2 can beseparated into oxidation to a lower oxidation state z+fl and subsequentoxidation to the oxidation state z+>2. The affinities of the particularoxidation processes are equivalent to the electrode potentials £2_0, £i-o>and E2-i'.

_ -AG 2_p

(z)F

( 3 - 1 2 3 )

181

It follows from the additivity of affinities, — AG2_() = — AG2_i — AG^Q, that

(z+,2 - z+>1)E§_i = z+,2^_0 - z+>1£?_o (3.2.24)

This relationship (sometimes called Luther's law) for the transfer of severalelectrons permits us to calculate one redox potential if the others areknown. Obviously, this is an analogy of the Hess law in thermodynamics.Equation (3.2.24) is not restricted to the case where the lowest oxidationstate is a metal.

Consider the reactions

A + e<^B and B + e<=±C (3.2.25)

The sum of these reactions yields a reaction termed disproportionation:

(3.2.26)aB

where K is the disproportionation constant. An inert electrode immersed ina solution containing species A, B and C will attain a potential correspond-ing to both of equilibria (3.2.25):

E = £°A,B + ^ l n ^ = Elc + ^ l n ^ (3.2.27)F aB F ac

If the system initially contains only form C with concentration c0, to whichthe oxidant is added, then the overall degree of oxidation is (3 = (2[A] +[B])/c0. If activities are set equal to concentrations, then the potential E canbe expressed in terms of j8:

+ * L / 3 l ± [ ( j 3 l ) + 4 j 3 ( 2 / W2F -j3 + 1 ± [03 - I)2 + 4)3(2 - j3)*:]1/2 l '

This dependence is shown in Fig. 3.10. The usual dependence for n = 2 isobtained for complete disproportionation, that is K » 1. With decreasing K,the E~P curve changes and at K = \ assumes the usual shape for n = 1. Forsmall K values three inflection points appear on the curve. This situationoften occurs for organic quinones. Form B is then termed the semiquinone.

A typical inorganic redox system of this type involves the equilibriumbetween metallic copper and copper(II) ions. In the absence of a complex-former the disproportionation constant is large and a solution of Cu+ ions isvery unstable. However, in the presence of ammonia, copper(I) ions arebonded in a complex, constant K decreases and the E-(3 curve exhibitsthree inflection points.

182

-8

1.6 2.00.8 1.2Overall oxidation degree,/3

Fig. 3.10 Dependence of the potential function v = (2F/2.3RT) • [E -K£A,B + £B,C)] on the overall oxidation degree J3 for a system withdisproportionation. The semiquinone formation constant K' = l/K is indi-

cated at each curve

3.2.5 Organic redox electrodes

Quinhydrone, a solid-state associate of quinone and hydroquinone,decomposes in solution to its components. The quinhydrone electrode is anexample of more complex organic redox electrodes whose potential isaffected by the pH of the solution. If the quinone molecule is denoted as Oxand the hydroquinone molecule as H2Red, then the actual half-cell reaction

Ox + 2e *± Red2" (3.2.29)

is accompanied by reactions between the hydroquinone anion Red2" andthe solvent. Hydroquinone is a weak dibasic acid, with the dissociationconstants

(3.2.30)

[HRed ]

183

For the quinhydrone electrode, the equation for the electrode potential hasthe form

PT [Hvl

(3.2.31)

where EOr is the formal potential. The concentration of the anionof the reduced form is substituted from the equation for the dissociationconstants and from the equation for the overall concentration of hydro-quinone, cRed,

, = [H2Red] + [HRecT] + [Red2"]

so that [Red2"] = cRedA:;A:2/([H+]2 + K[[H+] + K[K'2) and

\n([H] + K[[H] + K[K£ (3.2.33)

As [Ox] = cRed for quinhydrone, the second term on the right-hand side ofthis equation equals zero and the first and third are combined in theconstant E^h. The fourth term is simplified in different pH ranges as follows:

(a) In an acid medium, [H + ] 2 » K\K'2 + K[[H% so that

fH (3.2.34)

(b) If K[[H+] »[H+]2 + K[K'2, then

£qh = * q h + ~\n K\ - ^ ^ P H (3.2.35)

It Lt(c) In a rather alkaline region K[K'2» K[[H+] + [H+]2, dissociation is

complete and the potential is independent of pH:RT

Eqb = E*u + — \nK[K2 (3.2.36)At

Thus a plot of the dependence of Eqh on the pH (Fig. 3.11) will yield acurve with three linear regions (the second of which is poorly defined), withslopes of 0.0591, 0.0295 and 0 (at 25°C). The intersections of each twoneighbouring extrapolated linear portions yield the pK[ and pA^ values, asfollows from comparison of Eqs (3.2.34) to (3.2.36).

Figure 3.12 gives examples of the dependence of the apparent formalpotential on the pH for other organic systems (see also p. 465).

184

I I I !

10 I 12pK, pK2 pH

Fig. 3.11 The dependence of the potential of the quinhydroneelectrode £qh(V) on pH. The straight line (1) corresponds to Eq.

(3.2.34), (2) to (3.2.35) and (3) to (3.2.36)

In cases where the potential of an inert electrode, in the presence of theredox system of interest, is established slowly, another redox system maythen be added as a 'mediator'. The latter system should react rapidly withthe former and quickly establish an equilibrium at the electrode. As a verysmall amount of the mediator is added, concentration changes in themeasured system compared to the original state can be neglected. Suitablemediators are Ce4+-Ce3+, methylene blue-leucoform, etc.

Standard redox potentials can be determined approximately from thetitration curves for suitably selected pairs of redox systems. However, thesecurves always yield only the difference between the standard potentials anda term containing the activity coefficients, i.e. the formal potential. Thelarge values of the terms containing the activity coefficients lead to aconsiderable difference between the formal potential and the standardpotential (of the order of tens of millivolts).

3.2.6 Electrode potentials in non -aqueous media

While the laws governing electrode potentials in non-aqueous media arebasically the same as for potentials in aqueous solutions, the standardizationin this case is not so simple. Two approaches can be adopted: either asuitable standard electrode can be selected for each medium (e.g. thehydrogen electrode for the protic medium, the bis-diphenyl chromium(II)/bis-diphenyl chromium(I) redox electrode for a wide range of organic

185

Fig. 3.12 The dependence on pH of the oxidation-reductionpotential for cOx = cRcd: (1) 6-dibromphenol indophenol, (2) Lauth'sviolet, (3) methylene blue, (4) ferricytochrome c/ferrocytochrome

c, (5) indigo-carmine

solvents or, for example, the chlorine electrode for some melts—cf. Section3.2.1), or, on the other hand, all the potentials can be related to theaqueous standard hydrogen electrode on the basis of a suitable convention.The first approach is based on the discussion in Section 3.1.5 and requiresno further explanation.

To decide whether a unified electrode potential scale is of someadvantage, consider cells (3.1.40) in both aqueous medium and in proticmedium s:

P t |H 2 |HCla ± ,w |AgCl |Ag |P tPt |H 2 |HClf l ± , s |AgCl |Ag|Pt

186

The EMFs of these cells are given by the relationships

RT— In a±,Hci(w)

(3.2.37)

£(w) = s s 2 - — In

v j in tf±,HCI(,SJ

(3.2.38)

It is assumed that the activities are based on the same concentration scaleand that limC|_>0 flf7c/ = 1. For 0±>Hci(s) =

0±,HCI(W), the difference betweenE(s) and E(w) is given by the relationship

t r,H, +AG t r ,c l-

where A G ^ ^ is the overall standard Gibbs energy of transfer of HC1 fromwater to the solvent s and A G ? ; ^ and AG^cr8 are the individual standardGibbs energies of transfer of H+ and Cl~ from water to the solvent s (Eq.1.4.35). It should be noted that, so far, the derivation has been concernedwith the transfer of a species between two pure solvents, so that in this case,the standard Gibbs transfer energy will be somewhat different from thevalues measured for distribution equilibria. This type of standard Gibbstransfer energy must be determined on the basis of the solubility of thegiven electrolyte, present as a solid or gaseous phase, in equilibrium witheach of the pair of solvents.

The derivation for equilibrium between a solution and a gaseous phase isbased on the Henry law constant kif defined on page 5. The standard Gibbsenergy of the transfer of HC1 from a solution into water is

= RT In ( ^ Y ^ \ (3.2.40)

For the transfer equilibrium between a solution and a solid phase of theelectrolyte BA it holds that

A G - T = RT in (%&) = 2RTln " • ^ W (3.2.41)\PBA(S) / mBA(s)r±(s)

where the PBA'S a r e the solubility products, the mBA's the solubilities andy±'s the mean activity coefficients in each of the two phases.

As separation of the standard Gibbs transfer energy of the electrolyte as awhole into the individual contributions for the anion and the cation can, of

187

course, not be carried out by a thermodynamic procedure, a suitableextrathermodynamic assumption must be selected.

A suitable extrathermodynamic approach is based on structural con-siderations. The oldest assumption of this type was based on the propertiesof the rubidium(I) ion, which has a large radius but low deformability. V.A. Pleskov assumed that its solvation energy is the same in all solvents, sothat the Galvani potential difference for the rubidium electrode (cf. Eq.3.1.21) is a constant independent of the solvent. A furtherassumption was the independence of the standard Galvani potential of theferricinium-ferrocene redox system (H. Strehlow) or the bis-diphenylchromium(II)-bis-diphenyl chromium(I) redox system (A. Rusina and G.Gritzner) of the medium.

While the validity of these assumptions has been criticized, that adoptedby A. J. Parker has received the widest acceptance, i.e. that tetraphenyl-arsonium tetraphenylborate (Ph4AsPh4B),

yields ions with the same standard Gibbs energy of solvation in the samemedium. In other words, the standard Gibbs energy for the transfer of thetetraphenylarsonium cation and of the tetraphenylborate anion between anarbitrary pair of solvents is always the same (called the TATB assumption),that is

The structure of the ions, where the bulky phenyl groups surround thecentral ion in a tetrahedron, lends validity to the assumption that theinteraction of the shell of the ions with the environment is van der Waals innature and identical for both ions, while the interaction of the ionic chargewith the environment can be described by the Born approximation (seeSection 1.2), leading to identical solvation energies for the anion and cation.

The determination of the standard Gibbs transfer energy for an arbitraryion and arbitrary pair of solvents is based on the determination of thesolubility product of Ph4AsPh4B in both solvents or on the determination ofthe distribution coefficient of Ph4AsPh4B between the two solvents. Theseexperimental data then yield the standard Gibbs energies for the individualions, Ph4As+ and Ph4B~. If, for example, the standard Gibbs energy is to bedetermined for the transfer of the arbitrary cation C+ between a pair ofsolvents, the experimental data are determined for the salt CPh4B and the

188

required quantity is found from the equation

PTJ- (3.2.43)

These quantities can be used to rearrange the equation for the EMF of acell in non-aqueous medium into the form of a difference between theelectrode potentials on the hydrogen scale for aqueous solutions withadditional terms.

Equations (3.2.37), (3.2.38) and (3.2.39) yield the potential of thesilver-silver chloride electrode in non-aqueous medium

^Agci/Ag,ci-(s) = — — -p — —In acr (s)r t

A / *0,w—*s n j 1

= £A8c,/Ag,cr(w) j ^ - - — In aa-(s) (3.2.44)r r

and, considering Eq. (3.1.56), the potential of the hydrogen electrode in thesame medium (/? = 1),

£H+/H2(S) = H2 ^ tlAL~ + — In aH+(s)t t

(3.2.45)I 1

Thus, these relationships can be used to define a pH scale for non-aqueous protic media, consistent with the pH scale for aqueous solutions.For standard hydrogen pressure, the potential of the hydrogen electrodedepends on the pH(s) according to the relationship

£H+/H2(S) = -2303RT/F pH(s) (3.2.46)

Equation (3.2.45) then gives

pH(s) = -log aH+(s) - 0.4343AG?r'JTTs (3.2.47)

The values of the standard Gibbs transfer energies for H+ then determinethe solvent affinity for protons.

3.2.7 Potentials at the interface of two immiscible electrolyte solutions

The procedure described in the preceding section can form a basis forunambiguous determination of the Galvani potential difference between twoimmiscible electrolyte solutions (the Nernst potential) considering Eqs(3.1.22) and (1.4.34), e.g. for univalent ions,

where the upper sign is valid for a cation and the lower sign for an anionand A(p° is the standard potential difference for the transfer of ion / from

189

Table 3.6 Standard Gibbs transfer energies and standardelectric potential differences of transfer

Ion

(a) Water-nitrobenzene

Li+

Na+

Ca2+

Sr2+

H+

Ba2+

K+

Rb+

Cs+

Me4N+

Bu4N+

Ph4As+

crBr~NO3-

rPicrateTetraphenylborate

(b) Water-dichloroethane

Me4N+

Bu4N+

Ph4As+

crBr"

rPicrate

(kJ-moP 1 )

-38.4-34.5-68.3-67.2-32.5-63.3-24.3-19.9-15.5-3 .424.235.9

-30.5-28.5-24.4-18.8

4.635.9

-17.621.835.1

-46.4-39.3-26.4-6 .7

(V)'

0.3890.3580.3540.3480.3370.3280.2520.2060.1610.035

-0.251-0.372-0.316-0.295-0.253-0.195

0.0480.372

0.182-0.225-0.364-0.481-0.408-0.273-0.069

phase /} to phase oc. The values of these quantities and the standard Gibbsenergies for the transfer of ions from nitrobenzene or 1,2-dichloroethaneinto water are listed in Table 3.6. The values of standard Gibbs transferenergies in the water/organic solvent give a quantitative meaning to theterms, hydrophilic and hydrophobic. For hydrophilic ions AG?r'J~^»0 andthe opposite relation is valid for hydrophobic ions. Sometimes the ions withtheir standard Gibbs transfer energies around zero are called'semihydrophobic'.

Consider a system of two solvents in contact in which a single electrolyteBA is dissolved, consisting of univalent ions. A distribution equilibrium isestablished between the two solutions. Because, in general, the solvationenergies of the anion and cation in the two phases are different so that theion with a certain charge has a greater tendency to pass into the secondphase than the ion of opposite charge, an electrical double layer appears at

190

the interface (see Section 4.5.3) and a Galvani potential difference is formedbetween the two phases, called the distribution potential. The value of thedistribution potential follows from Eq. (3.2.48). If the activities of thecation and anion in a given phase are set equal to one another andequations of this type calculated once for cations, and then for anions, areadded, it follows that

AG ' A G (3.2.49)

The distribution potential A£0distr is thus independent of the ionconcentration.

Potential differences at the interface between two immiscible electrolytesolutions (ITIES) are typical Galvani potential differences and cannot bemeasured directly. However, their existence follows from the properties ofthe electrical double layer at the ITIES (Section 4.5.3) and from the kineticsof charge transfer across the ITIES (Section 5.3.2). By means of potentialdifferences at the ITIES or at the aqueous electrolyte-solid electrolytephase boundary (Eq. 3.1.23), the phenomena occurring at the membranesof ion-selective electrodes (Section 6.3) can be explained.

References

Antelman, M. S., and F. J. Harris, Jr., The Encyclopedia of Chemical ElectrodePotentials, Plenum Press, New York, 1982.

Bard, A. J. (Ed.), Electrochemistry of Elements, Plenum Press, New York,individual volumes appear since 1974.

Bard, A. J., J. Jordan, and R. Parsons (Eds), Oxidation-Reduction Potentials inAqueous Solutions, Blackwell, Oxford, 1986.

Butler, J. N., Reference electrodes in aprotic solvents, AE, 7, 77 (1970).Clark, W. M., Oxidation Reduction Potentials of Organic Systems, Williams and

Wilkins, Baltimore, 1960.Ives, D. J., and G. J. Janz (Eds), Reference Electrodes, Theory and Practice,

Academic Press, New York, 1961.Karpfen, F. M., and J. E. B. Randies, Ionic equilibria and phase-boundary

potentials in oil-water systems, Trans. Faraday Soc, 49, 823 (1953).Koryta, J., Electrochemical polarisation of the interface of two immiscible elec-

trolyte solutions I, Electrochim. Acta, 24, 293 (1979); II, Electrochim. Acta, 29,445 (1984); III, Electrochim. Acta, 33, 189 (1988).

Laity, R. W., /. Chem. Educ, 39, 67 (1962).Latimer, W. M., Oxidation Potentials, Prentice-Hall, New York, 1952.Michaelis, L., Oxidation Reduction Potentials, Lippincott, Philadelphia, 1930.Michaelis, L., Occurrence and significance of semiquinone radicals, Ann. New York

Acad. ScL, 40,39(1940).Parker, A. J., Solvation of ions—enthalpies, entropies and free energies of transfer,

Electrochim. Acta, 21, 671 (1976).Vanysek, P., Electrochemistry at Liquid-Liquid Interfaces, Springer-Verlag, Berlin,

1985.Wawzonek, S., Potentiometry: oxidation reduction potentials, Techniques of

Chemistry, (Eds. A. Weissberger and B. W. Rossiter), Vol. I, Part Ha,Wiley-Interscience, New York, 1971.

191

3.3 Potentiometry

Potentiometry is used in the determination of various physicochemicalquantities and for quantitative analysis based on measurements of the EMFof galvanic cells. By means of the potentiometric method it is possible todetermine activity coefficients, pH values, dissociation constants and solu-bility products, the standard affinities of chemical reactions, in simple casestransport numbers, etc. In analytical chemistry, potentiometry is used fortitrations or for direct determination of ion activities.

3.3.1 The principle of measurement of the EMF

The EMF of a galvanic cell is a thermodynamic equilibrium quatity. Thus,the potential of a cell must be measured under equilibrium conditions, i.e.without current flow. The measured EMF must be compensated by a knownexternal potential difference. The measurement of the EMF of a cell is thusbased on determination of a potential difference that exactly compensatesthe measured potential difference so that no current passes. This is easilyachieved by the Poggendorf compensation method (see Fig. 3.13).

At present, potentiometry is performed primarily using electronic instru-ments with solid-state elements. The EMF can be measured by compensa-tion measurement using a transistor voltmeter, i.e. according to the schemein Fig. 3.13, where the galvanometer is replaced by an amplifier and meter.A second possibility is to measure the tiny current passed on connectingthe cell with a large external resistance. Typical instruments of this kindconsist of three parts. The first part is the input circuit, acting as animpedance transducer. Mostly MOSFET elements or capacity diodes areused here. The second part contains a power amplifier, permitting the use ofa meter with large energy consumption. Depending on whether the EMFmeasured by using an a.c. amplifier is modulated in the input circuit (by ad.c. amplifier), an electronic demodulation circuit must be connected in thesecond part to rectify the amplified alternating voltage. The third part isthe indicator with a digital display.

If a cell is to be used as a potential standard, then it must be preparedas simply as possible from chemicals readily available in the required purityand, in the absence of current passage, it must have a known, defined,constant EMF that is practically independent of temperature. In this casethe efficiency, power, etc., required for cells used as electrochemical powersources is of no importance. The electrodes of the standard cell must not bepolarizable by the currents passing through them when the measuring circuitis not exactly compensated.

The mostly used Weston cell consists of mercurous sulphate and cadmiumamalgam electrodes:

Hg | Hg2SO4(s) | 3CdSO4.8H2O(sat.) | Cd,Hg(12.5 weight % Cd) (3.3.1)

192

Fig. 3.13 Poggendorfs compensation circuit.A uniform resistance wire is connected at itsends (points a and b) to a stable voltage source(e.g. a storage battery). A part of this voltagebetween one end of the potentiometric wireand the sliding contact c is fed into the othercircuit containing the measured source ofvoltage Ex, the null instrument G and the key.If no current flows through G the electricalpotential difference compensates the EMF, Ex.The voltage fed to the potentiometric wire iscalibrated by means of a standard cell, usuallyWeston cell, Es by means of the same com-

pensation procedure

3.3.2 Measurement of pH

The potentiometric measurement of physicochemical quantities such asdissociation constants, activity coefficients and thus also pH is accompaniedby a basic problem, leading to complications that can be solved only ifcertain assumptions are accepted. Potentiometric measurements in cellswithout liquid junctions lead to mean activity or mean activity coefficientvalues (of an electrolyte), rather than the individual ionic values.

The EMF of the cell

Ag | AgCl(s) | buffer,KCl(m) | H2,Pt (3.3.2)

suitable for pH measurements is given by an expression containing themean activity

log aH3O+ = 0A343F/RT(E + - log (mcrycl-) (3.3.3)

The pH value obtained in this way is accompanied by an error resultingfrom the approximation used for the calculation of y c r . Although this errormay be small in dilute solutions, the pH values obtained in this manner arenot exactly equal to the values corresponding to the absolute definition of

193

the pH in Eq. (1.4.39). The pH values given by this absolute definitioncannot, in principle, be exactly measured.

For current practice, the described method of pH measurement is tootedious. Moreover, not hydrogen but glass electrodes are used for routinepH measurements (see Section 6.3). Then the expression for the EMF ofthe cell consisting of the glass and reference electrodes contains a constantterm from Eq. (6.3.10), in addition to the terms present in Eq. (3.3.3); thisterm must be obtained by calibration. Further, a term describing the liquidjunction potential between the reference electrode and the measuredsolution must also be included.

In view of these difficulties an operational definition of pH has beenintroduced (cf. Section 1.4.6). This definition is based on the concept thatthe pH is measured with a hydrogen electrode, combined to the measuringgalvanic cell with a suitable reference electrode. The reference electrodesolution is connected with the solution of the hydrogen electrode through asalt bridge with KC1 of at least 3.5 mol • kg"1 concentration. The hydrogenelectrode is immersed first into the test solution X with pH(X) and the EMFE(X) is measured, and then into a standard solution S of pH(S),corresponding to potential E(S). It is assumed that the dependence of E onthe pH is linear with the Nernst slope, 2.303RT/F. Under these conditions,

(3.3.4)

Now the origin of the scale must be defined, i.e. a pH value must beselected for a standard (as close as possible to the value expected on the basisof definition 1.4.46). A solution of potassium hydrogen phthalate with amolality of 0.05 mol-kg"1 has been selected as the reference value pHstandard (RVS).

The pH of the saturated solution of potassium hydrogen phthalatecontaining chloride molarity mcl- is given by the equation

p H = -0.4343F(£ + £Ag/AgC1) + log [mcl-ycr] (3.3.5)where E is the EMF of the cell (3.3.2). The first two terms on the right-handside can be measured. If the pH of the cell (3.3.2) is determined for variousvalues of m c r and if the sum of these two terms if plotted vs mcl-, a nearlylinear dependence is obtained which may easily be extrapolated to zeroconcentration of chloride ions. For the pH(RVS), the pH of the standardphthalate solution alone, we have then

pH(RVS) = - lim [0.4343F(E + EAg/AgC1)/JRr-logmCi-]+ lim log yc, .mci~—*0 mcl—>0

(3.3.6)

The value of lim log ycl- may not be put equal to zero, as the overall ionicstrength of the solution is not equal to zero, but it may be calculated usingthe Bates-Guggenheim equation (1.3.35). The values of pH(RVS) obtainedin this way are listed in Table 3.7.

194

Table 3.7 Values of pH(RVS) for the reference valuestandard of 0.05 mol • kg"1 potassium hydrogen phthalate at

various temperatures

°c

051015202530

pH(RVS)

4.0003.9983.9973.9984.0014.0054.011

°C

35374045505560

pH(RVS)

4.0184.0224.0274.0384.0504.0644.080

°C

65707580859095

pH(RVS)

4.0974.1164.1374.1594.1834.2104.240

For practical measurements, six further solutions were measured asprimary standards and fifteen additional solutions as operational standards(the difference between these two types of standards lies in the presence orabsence of a liquid junction; they need not be distinguished for routinemeasurements).

In practice, the pH is mostly measured with a glass electrode (seeSection 6.3), connected with a calomel electrode (see Section 3.2.2). Themeasuring system is calibrated by using a single standard S, with a pH(S)value lying as close as possible to the pH(X) value. The pH(X) value is thencalculated from £(S), E(X) and pH(S) by Eq. (3.3.4). It is preferable to usetwo standards Si and S2, selected so that p H ^ ) is smaller and pH(S2) largerthan pH(X) (both the pH(S) values should be as close to pH(X) aspossible). The value of pH(X) is then calculated from the usual formula forlinear interpolation:

(3.3.7)E(S2)-E(Sl)

Analogously to water, standards are measured for a mixture of methanoland water (50 per cent by weight) as well as for heavy water, pD =-loga(D3O+).

An operational approach to the determination of the acidity of solutionsin deuterium oxide (heavy water) was suggested by Glasoe and Long. Thisquantity, pD, is determined in a cell consisting of an aqueous (H2O) glasselectrode and a saturated aqueous calomel reference electrode on the basisof the equation

= pHpHmeter reading 0.4 (3.3.8)

where the subscript pHmeter reading denotes the pH value indicated on theconventional pHmeter.

Determination of the pH in non-aqueous solvents is discussed in Section3.2.7.

195

3.3.3 Measurement of activity coefficients

Mean activity coefficients can be measured potentiometrically, mostly in aconcentration cell with or without transfer. Consider, for example, the cell(with a non-aqueous electrolyte solution)

Ag | AgCl(s) | KCl(mi) | K,Hg | KCl(m2) | AgCl(s) | Ag

On passing a positive charge from the left to the right in the graphicalscheme of this cell, silver is oxidized to form silver chloride; potassiumpasses through the amalgam into the other solution. Here, silver chloride isreduced to metallic silver and chloride ions. The overall reaction is thetransfer of KC1 from a region of higher concentration to a region of lowerconcentration, so that the EMF of the cell is given by the equation

^ a - ^ (3.3.9)F a±,2

where a2± = aK+acr is the mean activity of KC1. On rearrangement we

obtain

2RT 2RT——\na±l=——\na±2 + E

r r

J T2T OUT

= —— In ra2 + In y± 2 + E (3.3.10)F F

The concentration of solution 1 is kept constant while E is measured fordifferent concentrations of solution 2. The expression (2RT/F) In m2 + E isplotted against ra2. The value of the ordinate at point ra2 = 0 yields theterm (2RT/F) lna±>1 as In y±)2 = 0 at this point. Once the value of a±l isknown, then Eq. (3.3.10) and the measured E values can be used tocalculate the actual mean activity of the electrolyte at an arbitraryconcentration.

3.3.4 Measurement of dissociation constants

The dissociation constants of acids and bases are determined eitherexactly, by means of a suitable cell without liquid junction and withoutmeasuring the pH directly, or approximately on the basis of a pHmeasurement in a cell with liquid junction, the potential of which is reducedto a minimum with the help of a salt bridge. In the former case we shall use,for example, the cell

Pt,H2(/?H2 = 1) I HA(m1),NaA(m2),NaCl(m3) | AgCl(s) | Ag

whose EMF is given by the expression (where H3O+ is replaced by H+ for

196

simplicity)

RT1 ,

- — in (aH+acr)(3.3.11)

The dissociation constant, KA, of the acid HA is given by the equation

„ 7H+7A-

7HA 7HA(3.3.12)

where the apparent dissociation constant KA can be found, for example,conductometrically. It holds for the individual concentrations that

mC\- — = mx — (mH+ —

so that

- mOH- =

m2

m1- 4- Kv//mH+(3.3.13)

If the activity aH+ is substituted from Eq. (3.3.11) into Eq. (3.3.10) then

rearrangement yields

:A (3.3.14)2.303RT m yA-

The concentration in the second term on the left-hand side of this equationis expressed in terms of the known analytical concentrations, mly m2 andm3, and of the concentration mH

+, calculated from the apparent dissociation

-4.755

"--4.760-

.1

-4.765 0.05

Fig. 3.14 An extrapolation graph for determination of the thermo-dynamic dissociation constant of acetic acid using Eq. (3.3.14)

197

constant using Eq. (3.3.13). A series of measurements of E is carried out fora single mxlm2 ratio and varying sodium chloride concentrations ra3. Thenthe expression on the left-hand side of Eq. (3.3.14) is plotted against V7 andthe dependence is extrapolated to /—»0. The intercept on the ordinate axisyields the value of — log KA (see Fig. 3.14). However, the value of E°AgCl/Ag

must be the same as that employed for standard pH measurements.

References

Bates, R. G., Determination of pH. Theory, Practice. John Wiley & Sons, NewYork, 1973.

Covington, A. K., R. G. Bates and R. A. Durst, Definition of pH scales, standardreference values, measurement of pH and related terminology, Pure AppL Chern.,57, 531 (1985).

Glasoe, P. K., and F. A. Long, Use of glass electrode to measure acidities indeuterium oxide, /. Phys. Chem., 64, 188 (1960).

Harned, H. S., and B. B. Owen, The Physical Chemistry of Electrolytic Solutions,Reinhold, New York, 1950.

Robinson, R. A., and R. H. Stokes, Electrolyte Solutions, Butterworths, London,1959.

Serjeant, E. P., Potentiometry and Potentiometric Titrations, John Wiley & Sons,New York, 1984.

Wawzonek, S., see page 190.

Chapter 4

The Electrical Double Layer

Metal-solution interfaces lend themselves to the exact study of the double layerbetter than other types because of the possibility of varying the potential differencebetween the phases without varying the composition of the solution. This is donethrough the use of a reference electrode and a potentiometer which fixes thepotential difference in question. In favourable cases there is a range of potentials forwhich a current does not flow across the interface in a system of this kind, theinterface being similar to a condenser of large specific capacity. The capacity of thiscondenser gives a fairly direct measure of the electronic charge on the metallicsurface, and this, in turn, leads to other information about the double layer. No suchconvenient and informative procedure is possible with other types of interfaces.

D. C. Grahame, 1947

4.1 General Properties

In the interphase the cohesion forces binding the individual particlestogether in the bulk of each condensed phase are significantly reduced.Particles that had a certain number of nearest neighbours in the bulk of thephase have a smaller number of such neighbours at the interface. However,particles from the other phase can also become new neighbours. Thischange in the equilibrium of forces affecting particles at the interface canlead to a new lateral force, termed the interfacial tension. In addition, theinterphase usually has different electrical properties than the bulk phase.The situation becomes relatively simple when the phase is electricallycharged. The free charge is then centred in the interphase. Orientation ofdipoles in the interphase can also lead to a change in the electricalproperties. Further charges can enter the interphase through adsorption ofions and/or dipoles. Excess charge in the interphase resulting from thepresence of ions, electrons and dipoles produces an electric field. The regionin which these charges are present is termed the electrical double layer.

The presence of electrical charge affects the interfacial tension in theinterphase. If one of the phases considered is a metal and the other is anelectrolyte solution, then the phenomena accompanying a change in theinterfacial tension are included under the term of electrocapillarity.

While the formation of an electrical double layer at interfaces is a generalphenomenon, the electrode-electrolyte solution interface will be considered

198

199

first. If the electrode has a charge of <2(m), then this charge is distributeduniformly over the interface with the solution. If part of the electrode is notimmersed in the solution and is in contact with the air, then the portion ofthe charge corresponding to the surface in contact with the air is negligible,as this interface has minute capacitance. The great majority of the charge islocated at the metal-solution interface.

Excess charge in the metal <2(m) must be compensated in the solution bya charge of the same magnitude but of opposite sign. This charge isattracted from the solution by electrostatic forces. The general relationship

e(m) + G(s) = 0 (4.1.1)

is then valid, where Q(m) is the charge corresponding to the surface area Aof the interface on the electrode side and Q(s) is the charge in that part ofthe double layer in the solution.

According to the oldest hypothesis, put forward by H. L. F. Helmholtz,the electrical double layer has the character of a plate condenser, whoseplates are represented by homogeneously distributed charge in the metaland ions of the opposite charge lying in a parallel plane in the solution at aminimal distance from the surface of the metal. According to modernconceptions, the electron cloud in the metal extends to a certain degree intothe region of the layer of solvent molecules in the immediate vicinity of themetal surface. In this layer, the dipoles of the solvent molecules areoriented to various degrees towards the metal surface. Ions can gather inthe solution portion of the interphase as a result of the electrostatic field ofthe electrode ('electrostatic adsorption'). They can also, however, beadsorbed specifically on the electrode through van der Waals, hydrophobicand chemical forces.

If they are less polar than the solvent molecules, uncharged moleculesfrom the solution can also be adsorbed and replace solvent molecules thatwere originally in direct contact with the electrode.

In the electrode-solution interphase, the adsorption of these substances isalso affected by the influence of the electric field in the double layer on theirdipoles. Substances that collect in the interphase as a result of forces otherthan electrostatic are termed surface-active substances or surfactants.

In the simple case of electrostatic attraction alone, electrolyte ions canapproach to a distance given by their primary solvation sheaths, where amonomolecular solvent layer remains between the electrode and thesolvated ions. The plane through the centres of the ions at maximumapproach under the influence of electrostatic forces is called the outerHelmholtz plane and the solution region between the outer Helmholtz planeand the electrode surface is called the Helmholtz or compact layer.Quantities related to the outer Helmholtz plane are mostly denoted bysymbols with the subscript 2.

However, electrostatic forces cannot retain ions at a minimal distancefrom the electrode, as thermal motion continually disperses ions to a greater

200

distance from the phase boundary. In this way the Gouy or diffuse layer isformed, i.e. the region between the outer Helmholtz plane and the bulk ofthe solution. When only electrostatic forces act on the ions, the entirecharge is concentrated in this diffuse layer (see Fig. 4.1).

The charge accumulated by specific adsorption is partly compensated by a

o<t)OO3

ooo

B 1OO

oooo

©rJ ©

©©

© 0

©

y 0©

© }©0 =

©

f• ©

0

. 0

© © 8

Fig. 4.1 Structure of the electric double layer and electric potential distributionat (A) a metal-electrolyte solution interface, (B) a semiconductor-electrolytesolution interface and (C) an interface of two immiscible electrolyte solutions(ITIES) in the absence of specific adsorption. The region between the electrodeand the outer Helmholtz plane (OHP, at the distance x2 from the electrode)contains a layer of oriented solvent molecules while in the Verwey and Niessen

model of ITIES (C) this layer is absent

201

change in the charge in the diffuse layer. A plane through the centres of theadsorbed ions is termed the inner Helmholtz plane (and quantities con-nected with this plane are denoted by the subscript 1).

In order to be able to derive the basic relationships between quantitiescharacterizing the electrical double layer, the concept of an ideal polarizedelectrode must be introduced. Reversible electrodes, considered in Chapter3, are characterized by a potential that, at equilibrium at a giventemperature and pressure, is unambiguously determined by the composi-tion of the solution and electrodes, i.e. by the appropriate activities. If acharge is passed through an ideal reversible electrode, processes occur thatimmediately restore the original equilibrium; in this sense, such an electrodeis an ideal non-polarizable electrode. In contrast, an ideal polarizedelectrode can assume an arbitrary potential difference versus a referenceelectrode by the application of an external voltage source and maintain thispotential difference even after the voltage source is disconnected. Similarlyto the EMF of a reversible galvanic cell, this potential is an equilibriumquantity, as will be demonstrated in Section 4.2. In contrast to a galvaniccell, the potential of an ideal polarized electrode can be arbitrarily changedthrough a change in its charge without disturbing the electrode from theequilibrium state. Thus this electrode has one degree of freedom more thana non-polarizable electrode whose potential is determined by the composi-tion of the solution. An ideal polarized electrode is equivalent to a perfectcondenser without leakage.

The difference between an ideal polarized and ideal non-polarizableelectrode is best demonstrated in terms of the original Lippmann definition.The surface charge density of an ideal polarized electrode o(m) is equal tothe charge, Qy that must be supplied to the electrode from a voltage sourceso that its original potential against a reference electrode is maintainedduring a reversible increase in its surface area by one area unit. Chargea(m) is, of course, a function of the potential of the ideal polarizedelectrode and can even equal zero. For an ideal non-polarizable electrode,no charge need be supplied from an external source during a reversibleincrease in its surface area. Clearly, an electrode whose potential isdetermined by the given ions or oxidation-reduction system in solution hasthe properties of a condenser (with leakage), as an electrical double layerwith a certain capacity is formed at its surface. Nonetheless, for a very slowincrease in the surface of an ideal non-polarizable electrode, the half-cellreaction! is a sufficient source of charge for loading the electrical doublelayer. The supply of charge from an external source is then unnecessary formaintenance of a constant electrode potential.

Real electrodes, at which at least one electrode reaction occurs or canoccur with finite velocity (see Chapter 5), exhibit behaviour between that of

fThis term is used deliberately, rather than 'electrode reaction' (Chapter 5), as a thermo-dynamically reversible process is involved.

202

ideal polarized and ideal non-polarizable electrodes, especially duringcurrent passage.

Another definition of an ideal polarized electrode is based on thepractical form of this electrode. At an ideal polarized electrode either noexchange of charged particles takes place between the electrode and thesolution or—if thermodynamically feasible—exchange occurs very slowly asa result of the large activation energy.

Grahame gives the mercury electrode in a 1 M KC1 solution as a suitableexample of a nearly ideal polarized electrode. At a potential of —0.556 V(versus the normal calomel electrode) the following reactions can occur:2Hg^Hg2

2+ + 2e, K+ + Hg + e*±K,Hg and 2Cr«±Cl2 + 2e. However,the concentration of mercury (I) ions in solution is equal to 10~36 mol • dm"3,that of potassium in the amalgam equals to 10~45 mol • dm"3 and the partialpressure of chlorine is 10"57Pa. Clearly, such minute quantities will notmeasurably influence the electrode potential. A further possible reaction,2H2O + 2e«±H2 + 2OH~, corresponding to a relatively large partial pres-sure of hydrogen of 1.6 Pa, practically does not occur at the given potentialbecause of the high overpotential of hydrogen. Thus, this electrode fulfilsalmost perfectly the condition that no transfer of charged particles occursbetween the electrode and the solution.

Under real conditions it is very difficult to fulfil the condition of constantelectrode charge after disconnecting the external source. For example, anegatively charged electrode is discharged by the reduction of traces ofimpurities such as metal ions or oxygen.

Electrical double layers are also characteristic of the semiconductor-electrolyte solution, solid electrolyte or insulator-electrolyte solution inter-face and for the interface between two immiscible electrolyte solutions(ITIES) (Section 4.5).

References

Adamson, A. W., Physical Chemistry of Surfaces, 2nd ed. Interscience, New York,1967.

Barlow, C. A., The electrical double layer, PCAT, IX A, 167 (1970).Blank, M. (Ed.), Electrical Double Layer in Biology, Plenum Press, New York,

1986.Butler, J. A. V., Electrical Phenomena at Interfaces, Butterworths, London, 1951.Davies, J. T., and E. K. Rideal, Interfacial Phenomena, 2nd ed. Academic Press,

New York, 1963.Delahay, P., Double Layer and Electrode Kinetics, John Wiley & Sons, New York,

1965.Frumkin, A. N., V. S. Bagotzky, Z. A. Iofa, and B. N. Kabanov, Kinetics of

Electrode Processs (in Russian), Publishing House of the Moscow State Univers-ity, Moscow, 1952.

Grahame, D. C , Electric double layer, Chem. Rev., 41, 441 (1947).Marynov, G. A., and R. R. Salem, Electrical Double Layer at Metal-Dilute Solution

Interface, Lecture Notes in Chemistry, Vol. 33, Springer-Verlag, Berlin, 1983.

203

Mohilner, D. ML, The electric double layer, in Electroanalytical Chemistry (Ed. A.J. Bard), Vol. 1, p. 241, M. Dekker, New York, 1966.

Parsons, R., Equilibrium properties of electrified interfaces, MAE, 1, 103 (1954).Parsons, R., The structure of electrical double layer and its influence on the rates of

electrode reactions, AE, 1, 1 (1961).Parsons, R., Faradaic and non-faradaic processes, AE, 7, 177 (1970).Payne, R., The electrical double layer in nonaqueous solutions, AE, 7, 1 (1970).Perkins, R. S., and T. N. Andersen, Potentials of zero charge of electrodes, MAE,

5, 203 (1969).Spaarnay, M. J., The Electrical Double Layer, The International Encyclopedia of

Physical Chemistry and Chemical Physics, Topic 14, Vol. 4, Pergamon Press,Oxford, 1972.

4.2 Electrocapillarity

The interfacial tension at an interface is a force acting on a unit length ofthe interface against an increase in the interface area. The region aroundthe interface in which interfacial tension is produced is very narrow.Tolman, Kirkwood and Buff state that this distance is only about 0.1-0.3 nm for the liquid-vapour interface, corresponding to a monolayer at thesurface of the liquid, still affecting the next nearest layer. In order to derivea relationship that would qualitatively characterize the formation ofinterfacial tension at the interface between two homogeneous single-component phases, the following cycle will be considered. First a freesurface area of A is formed in each of phases a and /?. These surfaces arethen brought into contact, forming the interphase s. The work required toseparate these two phases and to return the system to the original state is— yA, where y is the interfacial tension. Thus,

yA = AG(or) + AG(j8) - AG(s) (4.2.1)

where AG(ar) and AG(/J) are the Gibbs energies required for reversibleseparation (i.e. the formation of two free surfaces A) of the phases a and jSboth divided by two. AG(s) is the Gibbs energy required for interruptingthe contact between phases a and jS. The AGs are given by the equations

AG(j8) = (-K03) - n.08)] + et(fi)

AG(s) = [ec(s) + ev(s)]N(sM (4.2.2)

where no(a) and no((i) are the numbers of nearest neighbours that theparticle has in the bulk of phases a and jS, respectively, ns(a) and ns(fi) arethe numbers of nearest neighbours from phase a or /? that the particle has atthe interface, ec(a) and ec(/J) are the bond energies between this particleand its nearest neighbour, £r's are the energies required for rearrangementof the particle in the interphase in order to be able to form a bond to a

204

particle in the neighbouring phase, ew(a), ev(j8), and £v(s) are the changesin the thermal vibrational energies of the particle at the interface as a resultof formation of the free surface and contact between the phases, N(a) andN(p) are the numbers of particles per unit surface area for the free surfaceof phases a and /3, £c(s) is the bond energy between the particles of phasesa and jS in the interphase and N(s) is the number of such bonds per unitsurface area at the interphase. If we assume that N(a) = N((3) = N(s)9 thatspecies at the surface have only one free bond (n0-ns=l) and that ec» er,£v, then

(s) (4.2.3)

This expression can be roughly interpreted as the difference between theGibbs energy of adhesion of the two phases and the sum of the Gibbsenergies of cohesion for the two phases.

These considerations can also be used to derive the Dupre equation,where AG(s) is the Gibbs energy of adsorption of the solvent per unit areaof the metal surface:

AG(s) = y m / a +y 1 / a -y (4.2.4)

where ym/a is the interfacial tension of the metal-air interface, y1/a is theinterfacial tension of the solution-air interface and y is the interfacialtension of the metal-solution interface.

Now the relationship between the interfacial tension and the compositionof the two phases in contact will be analysed thermodynamically by usingthe approach of J. W. Gibbs.

The interphase can be considered as a particular phase s of thickness h.This phase differs from the homogeneous phases only in that the effect ofpressure is accompanied by the effect of the interfacial tension y. Consider arectangle with sides h (perpendicular to the interphase) and / (parallel withthe interphase) located perpendicular to the interphase. The force acting onthe rectangle is not equal to the product phi (as for an area in the bulk ofthe solution) but phi — yl. If the volume of the interphase V(s) is increasedby dV(s) by increasing the thickness of the interphase by dh, then areaA = V(s)/h increases by cL4. The overall work, Wy connected with thisprocess consists of volume work accompanying the increase in the thicknessof the interphase and volume and surface work connected with an increaseof the surface area:

W(s) = -pA dh + {-ph + y) cL4 = -p dV(s) + y dA (4.2.5)

Because of this formation, a different definition of the enthalpy must beintroduced for the interphase, differing from the usual expression for ahomogeneous phase:

H(s) = U(s) + pV(s) - yA (4.2.6)

205

where U(s) is the internal energy of the interphase. For the differentialGibbs energy of the interphase we have

dG(s) = d[//(s) - TS(s)]= dU(s) + p dV(s) - V(s) dp - y dA - A dy

- T dS(s) - 5(s) dT + 2 ft d^i(s) (4-2.7)/=o

where nf-(s) is the amount of the ith component of the system in phase s andn is the total number of components. As

dU(s) + p dV(s) -ydA-T dS(s) = 0 (4.2.8)

it follows that

dG(s) = ~S(s) dT + V(s) dp - A dy + J) pg dnt(s) (4.2.9)/=0

The appropriate Gibbs-Duhem equation has the formn

-5(s) dT + V(s) dp - A dy - 2 n,-(s) d^ = 0 (4.2.10)i=0

Introduction of the surface concentrations of the components,

T * = ^ (4.2.11)

yields together with the relation V(s)/A = h

dy = - ^ dT + h dp - 2 r ; dju, (4.2.12)

This relationship is termed the Gibbs adsorption equation.It is often useful (e.g. for dilute solutions) to express the adsorption of

components with respect to a predominant component, e.g. the solvent.The component that prevails over m components is designated by thesubscript 0 and the case of constant temperature and pressure is considered.In the bulk of the solution, the Gibbs-Duhem equation, £, nt d/if = 0, isvalid, so that

- d j u 0 = E - d i u / (4.2.13);=i ^0

The sum in Eq. (4.2.12) can be modified using the relationship

i=0 1=1

o (4.2.14)n0 /

206

The surface excess of component / over component 0 is designated as F,:

F, = F * - — F£ (4.2.15)

and the Gibbs adsorption equation assumes the form

n

dy = - ^ r , d/i, (4.2.16)

In very dilute solutions, where n0 » niy F, ~ F*.When the adsorbed components are electrically charged, then the partial

molar Gibbs energy of the charged component depends on the charge of thegiven phase, and thus the chemical potentials in the above relationshipsmust be replaced by the electrochemical potentials. The Gibbs adsorptionisotherm then has the form

n

d y = - 2 rfdft (4.2.17)

The interfacial tension always depends on the potential of the idealpolarized electrode. In order to derive this dependence, consider a cellconsisting of an ideal polarized electrode of metal M and a referencenon-polarizable electrode of the second kind of the same metal covered witha sparingly soluble salt MA. Anion A~ is a component of the electrolyte inthe cell. The quantities related to the first electrode will be denoted as m,the quantities related to the reference electrode as m' and to the solution as1. For equilibrium between the electrons and ions M+ in the metal phase,Eq. (4.2.17) can be written in the form (s = n — 2)

dy = - F e d|Ue(m) - FM+ d ^ ^ m )s s

- 2 rf- djU/O) - F 2 I>, d<£(l) (4.2.18)i=i 1=1

The overall potential difference E and the potential of the referenceelectrode E(mf) are given by the equations

E = - — [jMe(m) - /ie(ni')] = EP ~ E(m')(4.2.19)

£(m') = E°(m') - ^ l n *A-(1) = £°(m') - \ foA-(0 " PA 0)]r r

where Ep is the electrode potential of the ideal polarized electrode on thehydrogen scale (this is a generalization of the electrode potential; seeSection 3.1.5).

The surface charge densities a(m) for the metal phase and a(l) for the

207

solution phase are expressed by the relationships

a(m) = (rM+ - Te)F* (4.2.20)

aQ) = -a(m) = Fjlzlrl1 = 1

It holds in the metal phase that jUM(m) = jUM+(m) 4- jUe(m) and in the salt

phase MA (denoted r) that //MAO) = /WO) + £A-(r). As fiM(m) and / / M AWare constants and because at equilibrium jUA-(r) = /iA-(l), jUM

+(r) = MM+(m')>it follows that

d£M+(m) + dAc(m) = 0

d£M+(m') + d/iA-(l) = < W 0 ) F d0(l) dA(m') = 0 l ' ' '

Combination of Eqs (4.2.20) and (4.2.21) with Eq. (4.2.18) yields

dy = ^ [d£c(m) - d£c(m') + djiA-(l)] - t r, d^-(l) (4.2.22)r /=i

It can further be seen from Eq. (4.2.19) that [djfte(m) - djfte(m')]/F =-dEp + dE(m') and also that d£(m') = -djuA (1)/F. The dependence of theinterfacial tension on the electrode potential is then given by the Gibbs-Lippmann equation,

dy = -a(m) d£p - £ T, d,u,(l) (4.2.23)i = l

The dependence of the interfacial tension on the potential is termed theelectrocapillary curve. It is convex to the axis of potential and oftenreminiscent of a parabola (see Fig. 4.2).

The first differential of this curve yields the surface charge densitya(m) = -a( l) :

-\HF) = a ( m ) (4-2-24)

p, T,u

The second differential of the electrocapillary curve yields the differentialcapacity of the electrode, C:

( _ C ( 4 2 2 5 )

This quantity is, in general, a function of the electrode potential.An important point on the electrocapillary curve is its maximum. It

follows from Eq. (4.2.24) that a(m) = a(\) = 0 at the potential of theelectrocapillary maximum.

This potential is termed the zero-charge potential and is denoted as Epzc.In earlier usage, this potential was also called the potential of theelectrocapillary zero; this designation is not suitable, as Epzc is connectedwith the zero charge a(m) rather than the zero potential.

208

3 6 0 -

-0.2 -0.4 -0.6 -0.8

Electrode potential vs. SCE.V

Fig. 4.2 Electrocapillary curves of 0.1 M aqueous solutions of KF, KC1, KBr,KI and K2SO4 obtained by means of the drop-time method (page 233). Theslight deviation of the right-hand branch of the SO4

2" curve is caused by ahigher charge number of sulphate. (By courtesy of L. Novotny)

Further, the concept of the integral capacity K will be introduced:

F — Fp P Z C

Ep — Epzc JEiCdE (4.2.26)

The differential capacity is given by the slope of the tangent to the curve ofthe dependence of the electrode charge on the potential, while the integralcapacity at a certain point on this dependence is given by the slope of theradius vector of this point drawn from the point Ep = Epzc.

209

The zero-charge potential is determined by a number of methods (seeSection 4.4). A general procedure is the determination of the differentialcapacity minimum which, at low electrolyte concentration, coincides withEpzc (Section 4.3.1). With liquid metals (Hg, Ga, amalgams, metals inmelts) Epzc is directly found from the electrocapillary curve.

As a material function, the zero-charge potential of various metals (andeven of different crystallographic faces of the same metal) extends over awide range of values (Table 4.1). How it is related to other materialfunctions like the electron work function 4>e has not yet been elucidated.However, S. Trasatti found two empirical equations, one for sp metals withthe exception of Ga and Zn (i.e. for Sb, Hg, Sn, Bi, In, Pb, Cd, Ti),£pzc(H2O) = 4>e/F - 4.69 V versus SHE, and the other one for the transi-tion metals (Ti, Ta, Nb, Co, Ni, Fe, Pd), Epzc = 3>e/F - 5.01 V versus SHE.

For solutions of simple electrolytes, the surface excess of ions can bedetermined by measuring the interfacial tension. Consider the valence-symmetrical electrolyte BA (z+ = —Z- = z). The Gibbs-Lippmann equationthen has the form

-dy = a(m) AE + FA d,uA (1) + I V <W(1) (4.2.27)while

a(l) = zF(IV " TA ) = a(m) (4.2.28)

Substitution of this equation into Eq. (4.2.27) and incorporation of therelationship djuB+(l) + djUA-(l) = RT d In (aB+aA-) = RT d In a\ yields

-dy = a(m)[d£p + jdfiA (1)] + rB

(4.2.29)

where EA is the potential of an ideal polarized electrode related to thepotential of a reference electrode reversible to anions A", EA = Ep — E^A/A

+ (RT/F) In aA-. For the surface excess FB+ we have

< 4 " 0 )

In this manner, the surface excess of ions can be found from theexperimental values of the interfacial tension determined for a number ofelectrolyte concentrations. These measurements require high precision andare often experimentally difficult. Thus, it is preferable to determine thesurface excess from the dependence of the differential capacity on theconcentration. By differentiating Eq. (4.2.30) with respect to EA and usingEqs (4.2.24) and (4.2.25) in turn we obtain the Gibbs-Lippmann equation

1 da(m) dFB+RTd\na2

± dEA'1 d2a(m) 1 dC d2FB+

RTdEAd\na2± RTd\na2

± 3E(4.2.31)

Table 4.1 Zero-charge potentials (vs. SCE). (According to A. N. Frumkin)

Electrode Epzc(y) Electrolyte Method of measurement

PbO2Au(llO)Pt(H)C(act.)CuSbHg

FeSnBi(ll l)Ag(ll l)Ag(100)Ag(llO)PbGaInTlCd

1.60-0.05-0.12-0.17-0.19-0.39-0.435

-0.61-0.62-0.66-0.695-0.870-0.975-0.80-0.85-0.89-0.95-0.99

5 mM H 2SO 4

5 mM NaF0.025 mMH2SO4

0.5 M Na2SO4 + 5 HIM H2SO4

O.lMNaOH2 mM KC1O4

All NaF

5 mM H 2SO 4

1 mM K2SO4

10 mM KF5 mM KPF6

5 mM KPF6

5 mM KPF6

1 mM NaFlMNaC104 + 0.1MHC104

3 mM NaF1 mM NaF1 mM NaF

Differential capacity minimumDifferential capacity minimumElectrokinetic potentialIon adsorptionContact (wetting) angleDifferential capacity minimumElectrocapillarityDifferential capacity minimumDifferential capacity minimumDifferential capacity minimumDifferential capacity minimumDifferential capacity minimumDifferential capacity minimumDifferential capacity minimumDifferential capacity minimumElectrocapillarityDifferential capacity minimumDifferential capacity minimumDifferential capacity minimum

Data for different Ag faces were taken from A. Hamelin, L. Stoicoviciu, L. Doubova, and S. Trasatti, Surface ScienceLetters, 201, L498 (1988).

211

Double integration with respect to EA yields the surface excess FB+;however, the calculation requires that the value of this excess be known,along with the value of the first differential dTB+/dEA for a definitepotential. This value can be found, for example, by measuring theinterfacial tension, especially at the potential of the electrocapillary maxi-mum. The surface excess is often found for solutions of the alkali metals onthe basis of the assumption that, at potentials sufficiently more negativethan the zero-charge potential, the electrode double layer has a diffusecharacter without specific adsorption of any component of the electrolyte.The theory of diffuse electrical double layer is then used to determine FB+and dTB+/dEA (see Section 4.3.1).

In practical measurements, the differential capacity values are determinedwith respect to a reference electrode connected to the studied electrolytethrough a salt bridge. The measured data are then recalculated for ananionic reference electrode by adding the value RT/F In a2

± to the Ep value.Figure 4.3 gives an example of the measured values of the surface excess foran electrolyte that is not adsorbed (KF) and that is adsorbed (KC1). It canbe seen that, at potentials Ep < -0.8 V (vs. NCE), the curves for the surfaceexcesses for both electrolytes merge, indicating that the two salts behaveidentically in this region of electrode potentials (because chloride ions areno longer adsorbed, the double layer is diffuse).

The quantity dy/d\na2± at the potential of the electrocapillary maximum

is of basic importance. As the surface charge of the electrode is here equalto zero, the electrostatic effect of the electrode on the ions ceases. Thus, ifno specific ion adsorption occurs, this differential quotient is equal to zeroand no surface excess of ions is formed at the electrode. This is especiallytrue for ions of the alkali metals and alkaline earths and, of the anions,fluoride at low concentrations and hydroxide. Sulphate, nitrate and perchlo-rate ions are very weakly surface active. The remaining ions decrease thesurface tension at the maximum on the electrocapillary curve to a greater orlesser degree.

In the case of a specific ('superequivalent') adsorption we have

where Fsalt is the surface excess for both components of the electrolyte. Oneof them may be specifically adsorbed and the second compensates thecorresponding excess charge by its excess charge of opposite sign in thediffuse layer.

References

Frumkin, A. N., Potentials of Zero Charge (in Russian), Nauka, Moscow, 1979.Frumkin, A. N., O. A. Petrii, and B. B. Damaskin, Potential of zero charge, CTE,

1, 221 (1980).

212

-20 -

-0.4 -0.8 -1.2Electrode potential ,V

-1.6

Fig. 4.3 Surface charge of the mercury electrode a(m) (JUC • cm 2) andsurface charges due to cations (FrB+) and anions ( F r A ) in 0.1 M KCl and

0.1 M KF at 25°C. (According to D. C. Grahame)

213

Parsons, R., Thermodynamic methods for the study of interfacial regions inelectrochemical systems, CTE, 1, 1 (1980).

4.3 Structure of the Electrical Double Layer

The thermodynamic theory of electrocapillarity considered above issimultaneously the thermodynamic theory of the electrical double layer andyields, in its framework, quantitative data on the double layer. However,further clarification of the properties of the double layer must be based on aconsideration of its structure.

At present it is impossible to formulate an exact theory of the structure ofthe electrical double layer, even in the simple case where no specificadsorption occurs. This is partly because of the lack of experimental data(e.g. on the permittivity in electric fields of up to 109 V • m"1) and partlybecause even the largest computers are incapable of carrying out such atask. The analysis of a system where an electrically charged metal in whichthe positions of the ions in the lattice are known (the situation is morecomplicated with liquid metals) is in contact with an electrolyte solutionshould include the effect of the electrical field on the permittivity of thesolvent, its structure and electrolyte ion concentrations in the vicinity of theinterface, and, at the same time, the effect of varying ion concentrations onthe structure and the permittivity of the solvent. Because of the unsolveddifficulties in the solution of this problem, simplifying models must beemployed: the electrical double layer is divided into three regions thatinteract only electrostatically, i.e. the electrode itself, the compact layer andthe diffuse layer.

On the basis of this model, the overall differential capacity C for a systemwithout specific adsorption, i.e. if the compact layer does not contain ions,is divided into two capacities in series, one corresponding to the compactlayer Cc and the other to the diffuse layer Cd:

It is assumed that the quantity Cc is not a function of the electrolyteconcentration c, and changes only with the charge a, while Cd depends bothon a and on c, according to the diffuse layer theory (see below). Thevalidity of this relationship is a necessary condition for the case where theadsorption of ions in the double layer is purely electrostatic in nature.Experiments have demonstrated that the concept of the electrical doublelayer without specific adsorption is applicable to a very limited number ofsystems. Specific adsorption apparently does not occur in LiF, NaF and KFsolutions (except at high concentrations, where anomalous phenomenaoccur). At potentials that are appropriately more negative than Epzc, whereadsorption of anions is absent, no specific adsorption occurs for the salts of

214

Li+, Na+ and K+ and the ions of some alkaline earths (cf. Fig. 4.3).

4.3.1 Diffuse electrical layer

The diffuse layer is formed, as mentioned above, through the interactionof the electrostatic field produced by the charge of the electrode, or, forspecific adsorption, by the charge of the ions in the compact layer. Inrigorous formulation of the problem, the theory of the diffuse layer shouldconsider:

(a) The individual ion dimensions(b) The effect of the electric field in the diffuse layer on the dielectric

properties of the solvent(c) The effect of the ions on the dielectric properties of the solvent

A rigorous solution of this problem was attempted, for example, in the'hard sphere approximation' by D. Henderson, L. Blum, and others. Herethe discussion will be limited to the classical Gouy-Chapman theory,describing conditions between the bulk of the solution and the outerHelmholtz plane and considering the ions as point charges and the solventas a structureless dielectric of permittivity e. The inner electrical potential0(1) of the bulk of the solution will be taken as zero and the potential in theouter Helmholtz plane will be denoted as 02- The space charge in thediffuse layer is given by the Poisson equation

d i v D = - / 9 (4.3.2)

where D is the electrical displacement given by the product of thepermittivity of the solution and of the electric field grad 0. If e is not afunction of the coordinates and a linear problem is involved (i.e. if in thediffuse layer the potential 0, charge density p and ion concentrations c, arefunctions only of the coordinate x perpendicular to the outer Helmholtzplane), then Eq. (4.3.2) becomes

d2d> pdiv grad 0 = —*f = - - (4.3.3)

ax e

The space charge density is given by the sum of the charges of all the ionspresent:

P = i(d)<t>ZiF (4.3.4)

where (c,)^ denotes concentration at potential 0. At equilibrium, theelectrochemical potential of any ion in any point of the solution is constant,(jU/) = (jU/)^=0- K, to a first approximation, the activities are replaced byconcentrations, then

**? + RT In (c,)*=o = tf + RT In (c,)* + ztF(t>

o exp(~^RT) (4'3-5)

215

Substitution for (c,)^ from Eq. (4.3.5) into Eq. (4.3.4) and then for p inEq. (4.3.3) yields the Poisson-Boltzmann differential equation

Z4P\ (4-3-6)dx e "<P=IJ

with the boundary conditions d0/d* = 0, (p = 0 for x—>o°. This differentialequation can be readily solved after multiplying both sides by d0/dx\Integration from x to infinity then gives (the concentration in the bulk of thesolution being denoted as ct rather than (c^^o)

while from the outer Helmholtz plane x = x2 to infinity

The Gauss theorem

(—) =— (4.3.8)

allows calculation of the charge ad in a column with unit cross-section in thediffuse layer:

(4.3.9)

If this equation is employed for a solution of a single valence-symmetricalelectrolyte {z+ = z_ = z)> then

(4.3.10)

ad = -V8£K7csinh ( ^ ) (4.3.11)

Thus, for water and a temperature of 25°C, after substitution for theconstants, when concentrations are expressed in moles per cubic decimetre,it follows that ad = 11.72c"2 sinh (19.46z02), [iC • cm"2. The quantity ad

expresses the total charge per square centimetre in the diffuse layer. If no

216

specific adsorption occurs, i.e. if no ions are present within the compact partof the double layer, then ad = a(l) = — a(m).

For small (f)2«RT/F, in Eq. (4.3.11) all the terms in the infinite seriesfor the hyperbolic sine except the first can be neglected, so that

(4.3.12)

where Lu = {eRTl2c)ml2F is the Debye length (see Eq. 1.3.17). Underthese conditions, the diffuse layer acts as a plate capacitor with charge ad

and thickness LD.The differential capacity of the diffuse layer is defined by the relationship

Cd= — dajd(t>2. According to this definition we obtain, from Eq. (4.3.11),

(4.3.13)

The total differential capacity can be either measured directly (see Section4.4) or calculated from Eq. (4.2.25). Because d[0(m) - 0(1)] = d£p, thedifferential capacity of the diffuse layer Cd can be calculated from Eq.(4.3.13), with substitution of the potential </)2 from Eq. (4.3.11). As for zerospecific adsorption qd = —q(m), the potential <f>2 is given by the relationship

2RT A o(m)" - sinrT1 ' v '

zF

RT<4-3-i4>

Figure 4.4 depicts the dependence of (f)2 on E - Epzc (the 'rationalpotential') for various electrolyte concentrations.

The charge density on the electrode a(m) is mostly found from Eq.(4.2.24) or (4.2.26) or measured directly (see Section 4.4). The differentialcapacity of the compact layer Cc can be calculated from Eq. (4.3.1) forknown values of C and Cd. It follows from experiments that the quantity Cc

for surface inactive electrolytes is a function of the potential applied to theelectrode, but is not a function of the concentration of the electrolyte. Thus,if the value of Cc is known for a single concentration, it can be used tocalculate the total differential capacity C at an arbitrary concentration ofthe surface-inactive electrolyte and the calculated values can be comparedwith experiment. This comparison is a test of the validity of the diffuse layertheory. Figure 4.5 provides examples of theoretical and experimentalcapacity curves for the non-adsorbing electrolyte NaF. Even at a concentra-tion of 0.916 mol • dm"3, the Cd value is not sufficient to permit us toset C « Cc.

217

-0.2

>

j-0.1

^ o

Diffu

se 1

1 1

• /

r ^JT

# i / i

1 com

i

i -—i

^— '

^

-

-

i i0.5 I -0 .5

Rational potential ,V

-1.0

Fig. 4.4 The dependence of the potential difference in thediffuse layer on the difference E — Epzc (the rational potential)for various concentrations of the surface inactive electrolyte

KF. (According to R. Parsons)

According to Eq. (4.3.13) the differential capacity of the diffuse layer Cd

has a minimum at 4>2 = 0, i-e. at E = Epzc. It follows from Eq. (4.3.1) andFig. 4.5 that the differential capacity of the diffuse layer Cd has a significanteffect on the value of the total differential capacity C at low electrolyteconcentrations. Under these conditions, a capacity minimum appears on theexperimentally measured C-E curve at E = Epzc. The value of Epzc can thusbe determined from the minimum of C at low electrolyte concentrations(millimolar or lower).

This theory of the diffuse layer is satisfactory up to a symmetricalelectrolyte concentration of 0.1mol-dm~3, as the Poisson-Boltzmannequation is valid only for dilute solutions. Similarly to the theory of strongelectrolytes, the Gouy-Chapman theory of the diffuse layer is more readilyapplicable to symmetrical rather than unsymmetrical electrolytes.

4.3.2 Compact electrical layer

The structure of the compact layer depends on whether specific adsorp-tion occurs (ions are present in the compact layer) or not (ions are absentfrom the compact layer). In the absence of specific adsorption, the surfaceof the electrode is covered by a monomolecular solvent layer. The solventmolecules are oriented and their dipoles are distorted at higher fieldstrengths. The permittivity of the solvent in this region is only anoperational quantity, with a value of about 12 at the Epzc in water,

-0.5 -1.0

Electrode potential^

-1.5 -2.0

Fig. 4.5 Dependence of the differential capacity C(JUF • cm"2) of a mercury electrode in 0 .916 M NaF (O) andin 1 mM NaF (•) on the electrode potential (vs. NCE). Thecurves correspond to calculations according to Eqs (4.3.1)and (4.3.13); curve 1 as Cc, curve 2 as C = (1/CC+ 1/Cd)~\

(According to D. C. Grahame)

219

decreasing to as little as 6 at potentials more distant from Epzc (for themercury-water interface; these values vary with the electrode metal).

It can be seen from Fig. 4.5 that a maximum (or 'hump') appears atpotentials somewhat more positive than the potential of the electrocapillarymaximum on the differential capacity-potential curve. This capacity maxi-mum can be explained by postulating that, in the corresponding potentialrange, the orientation of the water molecules is minimal, i.e. the permit-tivity is maximal; at more positive potentials, the water dipoles are orientedso that their negative ends (i.e. the oxygen atoms) are directed towards theelectrode surface while at more negative potentials this orientation isreversed. The disagreement between the potential of the 'hump' and thepotential of the electrocapillary maximum is apparently a result of weakchemical interaction between the water molecules and the surface of theelectrode. It should be noted that the dependence of the differentialcapacity for some mercury-solvent interfaces is simpler than that for water(e.g. for methanol or formic acid) and for others is more complicated(propylene carbonate, dimethylsulphoxide, sulpholane, etc.). In water, thehump disappears at higher temperatures.

The picture of the compact double layer is further complicated by the factthat the assumption that the electrons in the metal are present in a constantconcentration which discontinuously decreases to zero at the interface in thedirection towards the solution is too gross a simplification. Indeed,Kornyshev, Schmickler, and Vorotyntsev have pointed out that it isnecessary to assume that the electron distribution in the metal and itssurroundings can be represented by what is called a 'jellium': the positivemetal ions represent a fixed layer of positive charges, while the electronplasma spills over the interface into the compact layer, giving rise to asurface dipole. This surface dipole, together with the dipoles of the solventmolecules, produces the total capacity value of the compact double layer.

Specific adsorption occurs, i.e. ions enter the compact layer, in aconsiderable majority of cases. The most obvious result of specific adsorp-tion is a decrease and shift in the maximum of the electrocapillary curve tonegative values because of adsorption of anions (see Fig. 4.2) and topositive values for the adsorption of cations. A layer of ions is formed at theinterface only when specific adsorption occurs.

By means of the thermodynamic theory of the double layer and thetheory of the diffuse layer it is possible to determine the charge density ox

corresponding to the adsorbed ions, i.e. ions in the inner Helmholtz plane,and the potential of the outer Helmholtz plane 02 in the presence of specificadsorption.

The charge density of specifically adsorbed ions (assuming that anions areadsorbed specifically while cations are present only in the diffuse layer) for avalence-symmetrical electrolyte (z+ = —z_ = z) is

ax = -zF(TA - rA-,d) = -z /T A - f l (4.3.15)

220

where TA d is the surface anion concentration in the diffuse layer.Depending on its sign, the charge of the specifically adsorbed anions eitherincreases or decreases the charge of the metal part of the double layer,<j(m). The resultant charge, given by the sum ox 4- cr(m), affects the doublelayer electrostatically. As

CFJ + a(m) = -od (4.3.16)

Eq. (4.3.11) assumes the form

ox + a(m) = ySeRTc sinh —— (4.3.17)

\ 2.RT /

and Eq. (4.3.14) becomes

2RT (4-3-18>

The adsorbed ions (anions are considered) are located at the innerHelmholtz plane. In describing the distribution of the electrical potential inthe compact layer, the discrete structure of the charge in this layer must beconsidered, rather than continuous charge distribution, as was pointed outby O. A. Essin, B. V. Ershler, and D. C. Grahame. The inner Helmholtzplane is no longer an equipotential surface. As demonstrated by A. N.Frumkin, the potential at the centre of the adsorbed ion plays the greatestrole in the determination of the adsorption energy of ions. Ershler calledthis the micropotential. More exactly, it is the potential at the point wherethe given ion is adsorbed after subtracting the value of the coulombicpotential formed by this ion.

If the solution is considered to be a good electric conductor, then it isuseful to introduce the concept of electric image induced in an arbitraryconductor when a point charge is located close to its surface. The electricimage has the opposite charge with the same absolute value as the pointcharge and is situated inside the conductor at the same distance from thesurface as the original charge on the other side of the interface. The discretenegative charges induce a positive image charge in the solution, which isreflected on the outer Helmholtz plane (Fig. 4.6). The actual electric cloudof ions behind the outer Helmholtz plane is compressed so that it can bereplaced by this electric image.

The introduction of the concept of the micropotential permits derivationof various expressions for the potential difference produced by the adsorbedanions, i.e. for the potential difference between the electrode and thesolution during specific adsorption of ions. It has been found that, withsmall coverage of the surface by adsorbed species, the micropotentialdepends almost linearly on the distance from the surface. The distancebetween the inner and outer Helmholtz planes is denoted as xx_2 and thedistance between the surface of the metal and the outer Helmholtz plane asx2. The micropotential, i.e. the potential difference between the inner and

221

0++++

A1

x-O x1 x2

Fig. 4.6 The adsorbed anionA induces a positive chargeexcess at the outer Helmholtzplane, which can be repre-sented as its image A'. Theplane of imaging is identicalwith the outer Helmholtzplane. (According to B. B.Damaskin, O. A. Petrii and

V. V. Batrakov)

outer Helmholtz planes, (fri — <t>2, for the simple case of the electrocapillarymaximum, where there is no effect of the electrode charge, is givenapproximately by the equation

<t>\- 0 2 = - <t>2] (4.3.19)

In view of the assumed linear dependence of the electrical potential in thewhole compact layer, the above authors derived the following expression for0(m) — (j)2 when cr(m) = 0:

(4.3.20)

where ec is the permittivity in the compact layer, differing from thepermittivity in the bulk of the solution. For (j)1 — (f>2 and for (f>x alone the

222

substitution from Eq. (4.3.20) into Eq. (4.3.19) gives

( 4 - 3 - 2 1 )

X2

The adsorption of ions is determined by the potential of the innerHelmholtz plane <f)ly while the shift of Epzc to more negative values withincreasing concentration of adsorbed anions is identical with the shift in<t>(m). Thus, the electrocapillary maximum is shifted to more negativevalues on an increase in the anion concentration more rapidly than wouldfollow from earlier theories based on concepts of a continuously distributedcharge of adsorbed anions over the electrode surface (Stern, 1925). UnderStern's assumption, it would hold that <p(m) = <t>\ (where, of course, (pl nolonger has the significance of the potential at the inner Helmholtz plane).

According to Ershler, 0X must appear in the expression for theelectrochemical potential of ions adsorbed on the inner Helmholtz plane. Iftheir electrochemical potential is expressed by the equation

ft,! = tf + RT In TiA + A G ^ + z,F0, (4.3.22)

which is equal to the electrochemical potential for the same kind of ions inthe bulk of the solution (the activity of the ions in solution is at and thepotential 0(1) is again set equal to zero), then the equation for theadsorption isotherm is obtained:

^ ) ( ^ ) (4.3.23)

Here AGads is in the standard Gibbs energy change due to specificadsorption. To a first approximation, this quantity can be consideredindependent of the activity of the adsorbed ions; in general, AGads is afunction of the electrode charge and, at a given electrode charge (cr(m) = 0in the present case), is constant. The surface charge density ox for adsorbedanions with charge number z_ = — 1 is given by the expression

Ox = -FT-A = Ka- exp ( - ^ j (4.3.24)

Taking logarithms of Eq. (4.3.24) and differentiating with respect toyields

\3lnaJa(m) L+ RT\d In a J o(m)

!\ 1Ja,JUnaJ a ( m ) F L1 \d\na

223

From Eq. (4.3.26) the potential (j)1 at the electrocapillary maximum can becalculated:

F J L\ c m a _ / a ( m )

The integration constant is found from the condition that 0i = 0 2 for thelimiting value of the expression <91n |a1 | /31n«_->0 and c —>0 (see Eq.4.3.21).

If, for simplicity, 0 2 is neglected in Eqs (4.3.19) to (4.3.21) (valid forhigher electrolyte concentrations), then

0 i « — 0 ( m ) and a ^ — 0 ( m ) (4.3.28)

Substitution into Eq. (4.3.24) yields

0(jn) = —K a_ exp I 0 ( m ) ) (4.3.Zy)

where K' = x1_2K/ec. Hence

(2.303RT/F)x2/Xl_2 ^ ^

9 log a_ 1 - (/ir/F)(jc2/jc1_2)l/0(m)

As mentioned above, the quantity 0(m) is identified with the shift in thepotential of the electrocapillary maximum during adsorption of surface-active anions. For large values of this shift (0(m) »RT/F)

30(m) 2.303RT x2 ^ 2.303RT

F xx_2 zF

Indeed, for adsorption of iodide, Essin and Markov found a shift in theelectrocapillary maximum d(f)/d\oga_ of approximately —100 mV, inagreement with the theory.

At potentials far removed from the potential of zero charge, the electricalproperties of the compact layer are determined by both the charge of theadsorbed ions and the actual electrode charge. The simplest model for thissystem is one which assumes independent action of these two types ofcharge. The quantity 0(m) — 0 2 can then be separated into two parts,[0(m) — 02]a(m) and [0(m) —0 2 ] a i , each of which is a function of thecorresponding charge alone:

0(m) - 0 2 = [0(m) - 0 2 ] a ( m ) + [0(m) - 0 2 ] a i (4.3.32)

These two potential differences can both be expressed in terms of thecorresponding charges and two integral capacities:

0(m) - 02 = ,„ ,m + -£— (4.3.33)

224

It follows from the measurement that the quantity (/Qa(m) is a function ofthe charge <Hm)> but not of the concentration and the sort of theelectrolyte. The quantity (Kc)Ol changes with charges a(m) and ox and hasdifferent values for different electrolytes (70 \iV • cm"2 in iodide solutions,120 //F • cm"2 in chloride solutions).

Opinions differ on the nature of the metal-adsorbed anion bond forspecific adsorption. In all probability, a covalent bond similar to thatformed in salts of the given ion with the cation of the electrode metal is notformed. The behaviour of sulphide ions on an ideal polarized mercuryelectrode provides evidence for this conclusion. Sulphide ions are adsorbedfar more strongly than halide ions. The electrocapillary quantities (interfa-cial tension, differential capacity) change discontinuously at the potential atwhich HgS is formed. Thus, the bond of specifically adsorbed sulphide tomercury is different in nature from that in the HgS salt. Some authors havesuggested that specific adsorption is a result of partial charge transferbetween the adsorbed ions and the electrode.

4.3.3 Adsorption of electroneutral molecules

Electroneutral substances that are less polar than the solvent and alsothose that exhibit a tendency to interact chemically with the electrodesurface, e.g. substances containing sulphur (thiourea, etc.), are adsorbed onthe electrode. During adsorption, solvent molecules in the compact layerare replaced by molecules of the adsorbed substance, called surface-activesubstance (surfactant).t The effect of adsorption on the individual electro-capillary terms can best be expressed in terms of the difference of thesequantities for the original (base) electrolyte and for the same electrolyte inthe presence of surfactants. Figure 4.7 schematically depicts this depend-ence for the interfacial tension, surface electrode charge and differentialcapacity and also the dependence of the surface excess on the potential. Itcan be seen that, at sufficiently positive or negative potentials, the surfactantis completely desorbed from the electrode. The strong electric field leads toreplacement of the less polar particles of the surface-active substance bypolar solvent molecules. The desorption potentials are characterized bysharp peaks on the differential capacity curves.

For molecules with small dipoles, the adsorption region is distributedsymmetrically around the potential of the electrocapillary maximum.However, if chemisorption interaction occurs between one end of the dipole(e.g. sulphur in thiourea) and the electrode, the adsorption region is shiftedto the negative or positive side of the electrocapillary maximum.

The basic quantity in the study of adsorption is the surface excess of thesurface-active substance. In the formation of a monomolecular film of the

t Surfactants also include a number of ions with hydrophobic groups, such as tetraalkylam-monium ions with long-chain alkyl groups or dodecylsulphate.

15

-5

I

£

Fig. 4.7 Schematic dependence of the quantities y y°, a"°a, C" - C and T on the electrode potential. Thequantities with the superscript 0 refer to a surface-inactive electrolyte while those without a superscript refer

to an electrolyte containing a dissolved surfactant. (According to R. Parsons) toto

226

adsorbed substance, the maximum value of the surface excess Fmax isattained at complete coverage of the interface. The relative coverage ©, animportant term, is defined by the relationship

-I(4.3.34)

max

The fact that the electrocapillary quantities are measured relative to theirvalue in the base electrolyte can also be expressed in the formulation of theGibbs-Lippmann equation. If quantities referred to the base electrolyte areprimed and quantities referred to the studied surface-active substance aredenoted by the subscript 1, then

d(y' - y) = djr = -[a'(m) - a(m)] d£p

i = 2

; - r,) d^-(l) + I\ dMl(l) (4.3.35)

where Jt = yf-y is the surface pressure of the adsorbed surface-activesubstance. This is the force acting on a unit length in the direction of anincrease of the interface area, i.e. against the surface tension. The Gibbsadsorption isotherm can be derived from Eq. (4.3.65):

Tr-ip-) (4.3.36)

which acquires the following form for dilute solutions:

Equation (4.3.37) can be used to determine the function F1 = T^Cj), whichis the adsorption isotherm for the given surface-active substance. Substitu-tion for cx in the Gibbs adsorption isotherm and integration of thedifferential equation obtained yields the equation of state for a monomole-cular film Yx = T^Jt).

The simplest adsorption isotherm is that of Henry's law (linear adsorptionisotherm),

r1 = /8c1 (4.3.38)

where j3 is the adsorption coefficient. Substitution into Eq. (4.3.37) andintegration yields

Jt = RTT1 (4.3.39)

a two-dimensional analogy of the 'ideal gas law'.The limited number of free sites for the adsorbed substance on the

surface is considered by the Langmuir isotherm,

227

The basic assumption of the Langmuir adsorption isotherm is that theadsorbed molecules do not interact. This condition is not always fulfilled foradsorption, particularly on electrodes. The Frumkin adsorption isothermincludes interaction between molecules in the adsorption film,

0'ci=~exp(-«0) (4.3.41)

where the interaction coefficient a has a positive value for attractionbetween molecules and consequent favouring of adsorption, and a negativevalue for repulsion between molecules. It can be seen that the Langmuiradsorption isotherm is a special case of the Frumkin adsorption isotherm fora = 0, and for low surface coverage (0—>0) both are simplified into thelinear adsorption isotherm (see Fig. 4.8).

The adsorption coefficient /J or /?' is a function of the standard Gibbsadsorption energy alone when the molecules do not interact and theadsorption occurs without limitation, i.e. at very low electrode coverage.This dependence is expressed by the relationship

(4.3.42)

Fig. 4.8 The Frumkin adsorption isotherm. The value of theinteraction coefficient a is indicated at each curve. Thetransition in the metastable region is indicated by a dashed

line

228

The surface of a solid electrode is not homogeneous; even an apparentlysmooth surface as observed in an optical microscope contains corners andedges of the crystal structure of the metal and dislocations, i.e. sites wherethe regular crystal structure is disordered (see Section 5.5.5).

Adsorption on a physically heterogeneous surface differs from that on aphysically homogeneous surface. The heat of adsorption and Gibbs energyof adsorption differ from one site to another, and the sites with the lowestGibbs energy of adsorption are occupied first. M. Temkin derived anadsorption isotherm considering a physically heterogeneous surface. Thederivation was based on the assumption of a continuously heterogeneoussurface, where the free energy of adsorption per unit fraction dS of a unitsurface area increases linearly with increasing S (the unit surface is dividedinto elements dS, each of which is considered to be homogeneous):

dAGads=ordS (4.3.43)

where AGads is the value of the Gibbs adsorption energy for S = 0,A^ads + a for S = 1. If it is assumed that adsorption proceeds according tothe Langmuir isotherm on each homogeneous, planar element, then thecovered fraction of planar element dS is

T+7fc (4*344)

where

(4.3.45)

For the overall coverage 6 we have

0 = ffl(S)dS = — In o 1 + /?;C , , (4.3.46)

which is the basic form of the Temkin isotherm. In the central region of theisotherm, where /3{)c » 1»/30c exp (—a/RT)y a simpler form can be used,called the logarithmic isotherm,

RT0 = — \npoc (4.3.47)

The standard Gibbs energy of adsorption AG^s is mostly a function ofthe electrode potential. In the simplest model, the adsorption of a neutralsubstance can be conceived as the replacement of a dielectric with a largerdielectric constant (the solvent) by a dielectric with a smaller dielectric

229

constant (the surfactant) in a plate condenser, corresponding to theelectrode double layer. On the basis of this model, A. N. Frumkin and J. A.V. Butler derived a theoretical relationship for the dependence of AG^s onthe electrode potential:

AGSds = (AGads)max + a(Ep - £max)2 (4.3.48)

where £max is the potential at which the quantity -AGa d s has the maximumvalue (a >0). Substitution for -AG^ds into Eq. (4.3.42) yields

(4.3.49)

The appearance of peaks on the differential capacity curves can be derivedfrom this potential dependence in the following manner. The Gibbs-Lippmann equation (see Eq. 4.2.23) gives

( ) (4.3.50)

Differentiation with respect to Ep yields

Hence for the simplest case of a linear adsorption isotherm

rf = fji (4.3.52)Integration from cx = 0 to cx = c gives

(4.3.53)

where C" is the differential capacity of the electrode for cx = 0. Thedependence of capacity C on the concentration and on the electrodepotential is determined by the quantity d2fildE2

p, which, considering Eq.(4.3.49), is given by the relationship

d2p/3E2p = 2apo/(RT)[2a(Ep - Em)2/(RT) - 1] exp [-a(Ep - Emax)

2/(RT)](4.3.54)

Thus, the dependence of the differential capacity on the potential in thiscase has a minimum and two maxima. The potential minimum has a valueof Ep = Emax. As d2f}/dEl<0 in the vicinity of the minimum, it followsfrom Eq. (4.3.53) that the differential capacity decreases with increasingconcentration c in this region. The potentials of the maxima, given by therelationship Ep = £max ± ^3RT/2a, lie symmetrically on either side of £max.A qualitatively similar picture would be obtained for a more complexisotherm (see Fig. 4.9), but the peak potentials would depend on thesurfactant concentration.

230

1.0

Electrode potential^

1.5

Fig. 4.9 Dependence of the differential capacity of a mercury electrodeon the electrode potential Ep (vs. NCE) in a solution of 0.5 M NaF.Concentrations of f-C5HnOH: (1) 0, (2) 0.01, (3) 0.02, (4)

0.04 mol • dm 3 . (According to B. B. Damaskin)

The above relationships were derived for low electrode coverages by theadsorbed substance, where a linear adsorption isotherm could be used.Higher electrode coverages are connected with a marked change in thesurface charge. The two-parallel capacitor model proposed by Frumkin anddescribed by the equation

a = a o ( l - e ) + a 1 e (4.3.55)

describes this situation. Here a0 is the surface charge in the absence of the

231

surfactant and ox is the surface charge for complete coverage of theelectrode by the surfactant.

The described adsorption phenomena are characteristic for electrodes ofsp metals. Transition metal electrodes are usually connected with irrevers-ible chemisorption phenomena, discussed in Section 5.7.

References

Barlow, C. A., and J. R. MacDonald, Theory of discreteness of charge effects in theelectrolyte compact double layer, AE, 6, 1 (1967).

Damaskin, B. B., and V. E. Kazarinov, The adsorption of organic molecules, CTE,1, 353 (1980).

Damaskin, B. B., O. A. Petrii, and V. V. Batrakov, Adsorption of OrganicCompounds on Electrodes, Plenum Press, New York, 1971.

Frumkin, A. N., and B. B. Damaskin, Adsorption of organic compounds atelectrodes, MAE, 3, 149 (1964).

Grahame, D. C , see page 203.Henderson, D., L. Blum, and M. Lozada-Cassou, The statistical mechanics of the

electrical double layer, /. Electroanal. Chem., 150, 291 (1983).Kornyshev, A. A., W. Schmickler, and M. A. Vorotyntsev, Phys. Rev. B, 25, 5244

(1982).Marynov, A., and R. R. Salem, Electrical Double Layer at a Metal-Dilute

Electrolyte Interface, Vol. 33, Lecture Notes in Chemistry, Springer-Verlag,Berlin, 1983.

Mittal, K. L. (Ed.), Surfactants in Solution, Vols 1-3, Plenum Press, New York,1984.

Parsons, R., Structural effect on adsorption at the solid metal/electrolyte interface,J. Electroanal. Chem., 150, 51 (1983).

Parsons, R., Inner layer structure and the adsorption of organic compounds at metalelectrodes, 7. Electroanal. Chem., 29, 1563 (1984).

Reeves, R., The double layer in the absence of specific adsorption, CTE, 1, 83(1980).

Schmickler, W., A jellium-dipole model for the double layer, J. Electroanal. Chem.,150, 19 (1983).

Vetter, K. J., and J. W. Schultze, Potential dependence of electrosorption equilibriaand the electrosorption valence (in German), Ber. Bunsenges., 76, 920 (1972).

Void, R. D., and M. J. Void, Colloid and Interface Chemistry, Addison-WesleyPublishing Company, Reading, Mass., 1983.

Vorotyntsev, M. A., and A. A. Kornyshev, Models for description of collectiveproperties of the metal/electrolyte contact in the electrical double-layer theory,Elektrokhimiya, 20, 3 (1984).

4.4 Methods of the Electrical Double-layer Study

Of the quantities connected with the electrical double layer, the interfa-cial tension y, the potential of the electrocapillary maximum £pzc, thedifferential capacity C of the double layer and the surface charge densityq(m) can be measured directly. The latter quantity can be measured only inextremely pure solutions. The great majority of measurements has beencarried out at mercury electrodes.

Fig. 4.10 Capillary electrometer. The basiccomponent is the cell consisting of an ideallypolarized electrode (formed by the mercurymeniscus M in a conical capillary) and thereference electrode R. This system is connectedto a voltage source S. The change of interfacialtension is compensated by shifting the mercuryreservoir H so that the meniscus always has aconstant position. The distance between theupper level in the tube and the meniscus h ismeasured by means of a cathetometer C. (By

courtesy of L. Novotny)

233

G. Lippmann introduced the capillary electrometer to measure the surfacetension of mercury (Fig. 4.10). A slightly conical capillary filled withmercury under pressure from a mercury column (or from a pressurized gas)is immersed in a vessel containing the test solution. The weight of themercury column of height h is compensated by the surface tension accordingto the Laplace equation

p=^~hpHgg (4.4.1)

where p is the hydrostatic pressure at the mercury meniscus, r is the radiusof the capillary, pH g is the density of mercury and g is the standardacceleration of gravity. Both the polarized electrode and the referenceelectrodes are connected to a suitable external voltage source. The preciseposition of the meniscus in the capillary is adjusted by using a microscope.If the potential is changed, the interfacial tension of the electrode alsochanges and the original position of the meniscus is restored by the changein height h.

The dropping electrode method was suggested by B. Kucera. Thearrangement is similar to the Lippmann method, but the height of thecolumn h is sufficiently great for the mercury to drop from the end of thecapillary. The mercury drops are then collected and weighed or, as the flowrate of mercury through the capillary may easily be determined, thedrop-time is measured. Prior to dropping off, the mercury drop at the tip ofthe capillary with radius r is held by the force Inry. It thus falls off when itsweight w in solution is such that it equals the force 2nry\

- — )g = 2jtry (4.4.2)PHg/

where pso, is the density of the solution. This equation is not exactly obeyedby the dropping electrode, but, nonetheless, Kucera's method can be usedas a relative method. The measuring apparatus must be calibrated by meansof a standard solution with a known y-E dependence.

The wetting (contact) angle method is used for solid surfaces. If a gasbubble sticks to a metal surface, then the individual interfacial tensions aredistributed as shown in Fig. 4.11. It holds at equilibrium that

7sg = Ysi + 7ig cos 6 (4.4.3)

where 6 is the wetting angle.The quantity ysl is a function of the electrode potential, the quantity ylg is

independent of the potential and ysg should not depend on the potential,provided that the surface is dry below the bubble. As there is always a traceof moisture below the bubble in this arrangement, the value of ysg changesslightly with potential, but far less than ysI. The further the electrodepotential is from the potential of the electrocapillary maximum, the more ysl

decreases, i.e. cos 6 increases and the wetting angle 6 decreases. Obviously,

234

Fig. 4.11 Interfacial tensions between asolid, liquid and gaseous phase, ygs, yls

and ygl. 6 denotes the wetting angle

this method cannot be used to find absolute ysl values, but is at least usefulfor determining Epzc.

The differential capacity can be measured primarily with a capacitybridge, as originally proposed by W. Wien (see Section 5.5.3). The firstprecise experiments with this method were carried out by M. Proskurnin andA. N. Frumkin. D. C. Grahame perfected the apparatus, which employed adropping mercury electrode located inside a spherical screen of platinizedplatinum. This platinum electrode has a high capacitance compared to amercury drop and thus does not affect the meaurement, as the twocapacitances are in series. The capacity component is measured for thissystem. As the flow rate of mercury is known, then the surface of theelectrode A (square centimetres) is known at each instant:

A = 0.S5m2/3t2/3 (4.4.4)

where m is the flow rate of mercury (g • s"1) and t is the time elapsed sincethe beginning of formation of the drop. The capacitance of the system isproportional to this surface area. The values of the components of thebridge have to reach a balance during the droptime, and the time from thebeginning of formation of the drop is measured, i.e. from detachment of theprevious drop up to bridge balancing.

The surface charge of the electrode can be determined directly, e.g. fromthe polarographic charging current. During the growth of the dropelectrode, its capacitance increases, as does its charge. The charge that mustbe supplied to the electrode at constant potential produces the charging(capacity) current. The instantaneous value of this current is given, withreference to Eq. (4.4.4), as

, x cL4 da(m)7c=a(m) — + A — ^ (4.4.5)

In most inorganic electrolytes at constant potential cx(m) = constant,giving for the instantaneous electrode charge

<2(t) = f Ic df = 0.85m2/¥/3a(m) (4.4.6)Jo

The potential of the electrocapillary maximum can be found from theelectrocapillary curve while the Paschen method is an alternative. In this

235

case a mercury jet is injected into the solution, which has been thoroughlypurified and deoxygenated. In the absence of a sufficiently fast electrodereaction the potential of this mercury electrode indicates Epzc. The Paschenmethod is especially suitable for dilute solutions.

The surface excesses of solution components can be determined fromelectrocapillary measurements using the Gibbs-Lippmann equation (seeEq. 4.2.23) or the Gibbs adsorption isotherm (Eq. 4.3.36). Non-thermodynamic methods have been proposed for study of surfactantadsorption, especially of high-molecular-weight organic substances, basedon Frumkin's two-parallel capacitor model. Differentiating Eq. (4.3.55) withrespect to Ep gives

C = C0(l - O) + C,® + (a, - a0) ^ - (4.4.7)d£p

where Co is the differential capacity in the base electrolyte alone and Cx isthe differential capacity for complete electrode coverage by the surfactant.As we can set d0/d£ p ~O at potentials sufficiently distant from thedesorption potential, the relative electrode coverage is then

(4.4.8)

Other methods, based on kinetic methods of measuring surface excesses,will be discussed in Section 5.7.

References

Bard, A. J., and L. R. Faulkner, Electrochemical Methods, John Wiley & Sons,New York, 1980.

Grahame, D. C , see page 202.Sluyters-Rehbach, M., and J. H. Sluyters, A.C. techniques, CTE, 9, 177 (1984).Southampton Electrochemistry Group, Instrumental Methods in Electrochemistry,

Ellis Horwood, Chichester, 1985.

4.5 The Electrical Double Layer at the Electrolyte-Non-metallic PhaseInterface

4.5.1 Semiconductor-electrolyte interfaces

The interfaces between a semiconductor and another semiconductor (e.g.the very important pin junction, the interface between p- and n-typesemiconductors), between a semiconductor and a metal (the Schottkybarrier) and between a semiconductor and an electrolyte are the subject ofsolid-state physics, using a nomenclature different from electrochemicalterminology.

236

The basic difference between metal-electrolyte and semiconductor-electrolyte interfaces lies primarily in the fact that the concentration ofcharge carriers is very low in semiconductors (see Section 2.4.1). For thisreason and also because the permittivity of a semiconductor is limited, thesemiconductor part of the electrical double layer at the semiconductor-electrolyte interface has a marked diffuse character with Debye lengths ofthe order of 10~4-10"6cm. This layer is termed the space charge region insolid-state physics.

The description of the properties of this region is based on the solution ofthe Poisson equation (Eqs 4.3.2 and 4.3.3). For an intrinsic semiconductorwhere the only charge carriers are electrons and holes present in theconductivity or valence band, respectively, the result is given directly by Eq.(4.3.11) with the electrolyte concentration c replaced by the ratio n°/NA,where n°c is the concentration of electrons in 1 cm3 of the semiconductor in aregion without an electric field (in solid-state physics, concentrations areexpressed in terms of the number of particles per unit volume).

For doped semiconductors, it is assumed that the charge density in thesemiconductor, e.g. of type ny is

p(x) = e[ne(x)-n°] (4.5.1)

where ne(x) is the concentration of electrons and n°c is the electronconcentration in the absence of an electric field, practically equivalent to theconcentration of electron donors in the semiconductor. This is identical tothe concentration of positive ions, independent of the field in the semicon-ductor. It is assumed that the concentration of electron acceptors is muchlarger than the concentration of electron donors and of electrons and holesfor the original undoped intrinsic semiconductor.

It follows from Eq. (4.3.5) that

(4.5.2)

where <j)sc is the inner potential of the bulk of the semiconductor. Theprocedure on page 214 can be used to yield

(4.5.3)

Equation (4.5.3) gives, for (j>«kT/e with respect to Eqs (4.3.8) and(4.3.12)

d<l>/dx = [ekT/(2e2n°e)]m(<t> - 0SC) = ( 0 - 0SC)/LSC (4.5.4)

where Lsc is the Debye length in the semiconductor.In view of Eqs (4.3.8) and (4.5.3), where 0 — 0SC is replaced by the

overall potential difference between the bulk semiconductor and the

237

interface, 0SC - 0S = Af0, we find for the surface charge of the semicon-ductor

^ ^ ^ f (4.5.5)

The overall Galvani potential difference between the bulk of thesemiconductor and the bulk of the electrolyte solution can be separated intothree parts:

Ar<£ = Af 0 + A|0 + Al2(j) (4.5.6)

where Af$ is the potential difference between the bulk of the semiconduc-tor and the semiconductor-electrolyte interface, A|0 is the potentialdifference in the compact layer (i.e. between the interface and the outerHelmholtz plane) and A?0 is the potential difference between the outerHelmholtz plane and the bulk of the electrolyte solution. Obviously,

(4.5.7)

where £sc, Ex and Ex are the electric field strengths in the semiconductor, inthe compact layer and in the diffuse layer in the electrolyte, respectively, Lx

is the Debye length in the solution, dx is the thickness of the compact layerand £sc, £H

a nd £i are the permittivities of the semiconductor, compact layerand electrolyte, respectively. At sufficiently large electrolyte concentrations,the second and third terms in parentheses in Eq. (4.5.7) can be neglected.Under these conditions, the Galvani potential difference between thesemiconductor and the electrolyte solution is given by the potentialdifference in the space charge region (diffuse layer) in the semiconductor,which also holds for larger potential differences than RTIF.

The distribution of charge carriers in the space charge region of an ft-typesemiconductor is shown in Fig. 4.1. It is assumed that the semiconductorcontains not only electrons (majority charge carriers) but also much lowerconcentrations of holes (minority charge carriers). If the negative potentialof an ft-type semiconductor electrode increases, then the negative charge ofthe electrode increases through an increase in the concentration of electronsin the interphase (accumulation layer). As the electron energy in the spacecharge region e(p(x) depends on the distance from the interface, the energybands bend downwards in the direction towards the interface in thenegatively charged space charge region (Fig. 4.12A).

When the surface charge decreases to zero, the energy bands becomehorizontal. The corresponding flat-band potential A* 0 ^ is an analogy of thezero-charge potential Epzc (Fig. 4.12B).

When the difference A^C0 - A* 0^ becomes positive, the interphase isdepleted of majority charge carriers (electrons in this case) forming a

00

n°.nD

SCs

I"•i"S

1

sc

1

SC

"L

«

•g

Fig. 4.12 Dependence of concentrations of negative charge carriers («e) and positive charge carriers («p) on distance from theinterface between the semiconductor (sc) and the electrolyte solution (1) in an «-type semiconductor. These concentrationdistributions markedly differ if the semiconductor/electrolyte potential difference AfQ is (A) smaller than the flat-bandpotential Ar0 f b , (B) equal to the flat-band potential, (C) larger and (D) much larger than the flat-band potential. nD denotes

the donor concentration. (According to Yu. V. Pleskov)

239

depletion or Mott-Schottky layer (Fig. 4.12C). However, Eq. (4.5.5) is asimplification, as it neglects the presence of minority charge carriers, whichwas originally very low. The following relationship holds between theconcentration of electrons and the concentration of holes np:

npnc = nl (4.5.8)

where n0 is the concentration of electrons in the same semiconductor in theabsence of impurities and of an electric field inside the semiconductor, i.e.in the corresponding intrinsic semiconductor. According to the quantumtheory of semiconductors, the quantity n0 is given by the relationship

n0 = -3 exp

where me is the electron mass, h is the Planck constant and eg is the widthof the forbidden band.

For a sufficiently large potential increase, the charge in the interphasefinally corresponds to the minority charge carriers (Fig. 4.12D). The greaterthe width of the forbidden band eg, the broader is the potential range A\c<j>in which the space charge region has the character of a depletion layer, i.e.is formed by ionized impurity atoms.

It should be noted that the study of the properties of an electrical doublelayer at semiconductor electrodes corresponds to study of the cell (in thesimplest formulation)

M | S | electrolyte solution | M (4.5.10)

where M denotes the metal phase and S is the semiconductor. Theelectrolyte solution has a composition such that the interface with the metalis unpolarizable. An electrical double layer is formed at the interfacebetween the metal and the semiconductor (called the Schottky barrier), butthis interface is unpolarized, so that

Am0 = [Me(s) - jUe(m)]/e (4.5.11)

The discussion of Eq. (4.5.7) has shown that the potential of a polarizedsemiconductor electrode at sufficiently high electrolyte concentrations is

Ep « Af 0 + constant (4.5.12)

In Eq. (4.5.5), describing an rc-type semiconductor strongly doped withelectron donors, the first and third terms in brackets can be neglected forthe depletion layer (Alc(f)»kT/e). Thus, the Mott-Schottky equationis obtained for the depletion layer,

asc = Vlenle Af<j> (4.5.13)

The differential capacity of an Ai-type semiconductor electrode in the

240

Mott-Schottky approximation is

Q(JS

C(4.5.14)

The Mott-Schottky plot following from Eqs (4.5.12) and (4.5.14) is therelationship

2C"2 = —5- £ p + constant (4.5.15)

which can be used to determine the concentration of electron donors

It has so far been assumed that the semiconductor-electrolyte interphasedoes not contain either ions adsorbed specifically from the electrolyte orelectrons corresponding to an additional system of electron levels. Thesesurface states of electrons are formed either through adsorption (theShockley levels) or through defects in the crystal lattice of the semiconduc-tor (the Tamm levels). In this case—analogously as for specific adsorptionon metal electrodes—three capacitors in series cannot be used to charac-terize the semiconductor-electrolyte interphase system and Eq. (4.5.6) mustinclude a term describing the potential difference for surface states.

The presence of surface states results in a certain compression of thespace charge region and leads to a decrease in the band bending.

The situation of the electric double layer at a semiconductor/electrolytesolution interface affected by light radiation will be dealt with in Section5.10.

4.5.2 Interfaces between two electrolytes

The electrical double layer has also been investigated at the interfacebetween two immiscible electrolyte solutions and at the solid electrolyte-electrolyte solution interface. Under certain conditions, the interfacebetween two immiscible electrolyte solutions (ITIES) has the properties ofan ideally polarized interphase. The dissolved electrolyte must have thefollowing properties:

(a) The electrolyte in the aqueous phase, BXAX (assumed to dissociatecompletely to form cation B ^ and anion A ^ ) , is strongly hydrophilic,while the electrolyte in the organic phase, B2A2 (completely dissociatedto cation B2

+ and anion A2~), is strongly hydrophobic. These propertiescan be expressed quantitatively by the conditions for the distributioncoefficients between water and the organic phase k^°Ax and k^°Al:

< oA l « l , ^ A 2 » 1 (4.5.16)

(b) The equilibria of the exchange reactionsBi+(w) + B2

+(o) ^± Bj+(o) + B2+(w)

Ar(w) + A2-(o)«±Ar(o) + A2~(w) y - - )

241

are shifted to the left. In other words, the standard potential differencebetween water and the organic phase for the transfer of the cation ofthe aqueous phase and the anion of the organic phase must be aspositive as possible and the corresponding potential difference for thecation of the organic phase and the anion of the aqueous phase must beas negative as possible. In addition, there must be a sufficiently largedifference between the less positive value of A™ 0? for the first pair andthe less negative value for the second pair (cf. Section 3.2.8).

Under these conditions, the potential difference between the two phasesAQ0 can be changed by charge injection from an external electric currentsource. The appropriate experimental arrangement is shown in Fig. 5.18.

E. J. W. Verwey and K. F. Niessen described the electric double layer atITIES using a simplifying assumption that consists of only two diffuseelectric layers, each in one of the phases (see Fig. 4.1C). The overallpotential difference between the two phases, A^^, is thus given by therelationship

A:0 = 02(o)-02(w) (4.5.18)where the potential differences in the diffuse layers, 02(o) and 02(w), aredefined such that the potential in the bulk of the phase is subtracted fromthe potential at the interface. The terms <j)2{o) and $2(w) are given by Eq.(4.3.14), where the appropriate permittivity, e(w) or e(o), is substituted foreach phase, along with the electrolyte concentrations c(w) and c(o). As thesurface charges a(w) and a(o) are equal except for the sign (cf. Eq. 4.1.1),Eqs (4.5.18) and (4.3.14) can be used to calculate the dependence of $2(o)and 02(w) on A ^ . The electrical double layer has, of course, a similarstructure when a single electrolyte is present in the distribution equilibriumin the system.

It is interesting that the experimentally measured zero-charge potential ispractically identical with the value of A™ 0 = 0, calculated using the TATBassumption (3.2.64). This fact helps to justify the use of this assumption.

The electrical double layer at the solid electrolyte-electrolyte solutioninterface has been studied primarily in colloid suspensions, especially forsilver halides. The potential difference between the solid and liquid phasescan be changed by changing the concentration of the 'potential-determiningions', i.e. either the silver or halide ions. In solid oxide suspensions,hydrogen ions act as potential-determining ions. The zero-charge potentialof this system can be found from the dependence of the electrophoreticmobility (see Section 4.5.4) on the concentration of potential-determiningions, i.e. it corresponds to zero electrophoretic mobility. With this type ofinterface the structure of the electrical double layer depends to a markeddegree on the preparation and thus also on the final structure of the solidphase. Two cases are most often observed:

(a) Ions are adsorbed from solution on the surface of the solid phase andcounter ions form a diffuse layer.

242

(b) The electrical double layer has a structure similar to ITIES, but thediffuse layer in the solid becomes a simple Helmholtz layer because ofthe high concentration of ions.

It should, however, be noted that the electrical double layer at themetal-fused electrolyte interface does not have this character, in spite of theion concentration being high. In this system, the space charge includesseveral ion layers at the interface.

4.5.3 Electrokinetic phenomena

The adsorption of ions at insulator surfaces or ionization of surfacegroups can lead to the formation of an electrical double layer with thediffuse layer present in solution. The ions contained in the diffuse layer aremobile while the layer of adsorbed ions is immobile. The presence of thismobile space charge is the source of the electrokinetic phenomena.^Electrokinetic phenomena are typical for insulator systems or for a poorlyconductive electrolyte containing a suspension or an emulsion, but they canalso occur at metal-electrolyte solution interfaces.

Consider a solid surface in contact with a dilute electrolyte solution. Theplane where motion of the liquid can commence is parallel to the outerHelmholtz plane but shifted in the direction into the bulk of the solution.The electric potential in this plane with respect to the solution is termed theelectrokinetic potential £(|£| ^ |$2|)«

Of the four electrokinetic phenomena, two (electroosmotic flow and thestreaming potential) fall into the region of membrane phenomena and willthus be considered in Chapter 6. This section will deal with the electrophor-esis and sedimentation potentials.

If the electric field E is applied to a system of colloidal particles in aclosed cuvette where no streaming of the liquid can occur, the particleswill move with velocity v. This phenomenon is termed electrophoresis. Theforce acting on a spherical colloidal particle with radius r in the electric fieldE is 4^rerE02 (for simplicity, the potential in the diffuse electric layer isidentified with the electrokinetic potential). The resistance of the medium isgiven by the Stokes equation (2.6.2) and equals 6jrr/rv. At a steady state ofmotion these two forces are equal and, to a first approximation, theelectrophoretic mobility v/E is

In closer approximations, correction must be made for conductivity effects(relaxation and electrophoretic) and for the real shape of the particles. Thus,the velocity of electrophoretic motion depends on the composition of the

t This term must be distinguished from the concept of electrochemical kinetics, discussed inChapter 5.

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solution, on the properties of the surface of the particles and also on thecharge of the particles themselves. If ampholytic particles are involved, thenit also depends markedly on the pH, as the particles obtain a charge throughdissociation that is dependent on the pH. This fact was utilized by A.Tiselius to develop an electrophoretic method that is especially useful forproteins. At a given pH various proteins are ionized to a different degreeand also have different mobilities. The original single sharp boundarybetween a solution of a protein mixture in a suitable buffer and the purebuffer separates into several boundaries in an electric field, corresponding todifferently mobile components. Thus electrophoresis permits analysis of amixture of proteins without destructive chemical reactions. The experimen-tal methods for determining the boundary position in either classical or freeelectrophoresis are the same as in the study of diffusion (see Section 2.5.5).In addition, electrophoresis can be carried out by saturating a suitableporous carrier, e.g. filter paper, with a pure buffer and applying the studiedsolution as spots or bands. The evaluation methods are analogous to thoseused in paper chromatography. Electrophoresis can be used preparatively toseparate the components of a mixture, to concentrate fine suspensions insolution, etc.

A further electrokinetic phenomenon is the inverse of the formeraccording to the Le Chatelier-Brown principle: if motion occurs under theinfluence of an electric field, then an electric field must be formed by motion(in the presence of an electrokinetic potential). During the motion ofparticles bearing an electrical double layer in an electrolyte solution (e.g. asa result of a gravitational or centrifugal field), a potential difference isformed between the top and the bottom of the solution, called thesedimentation potential.

References

Andrews, A. T., Electrophoresis, Theory, Techniques and Biochemical and ClinicalApplications, Oxford University Press, Oxford, 1982.

Boguslavsky, L. I., Insulator/electrolyte interface, CTE, 1, 329 (1980).Butler, J. A. V., see page 202.Gaal, O., G. Medgyesi, and L. Vereczkey, Electrophoresis in the Separation of

Biological Macromolecules, John Wiley & Sons, New York, 1980.Gerischer, H., Semiconductor electrode reactions, AE, 1, 31 (1961).Gerischer, H., Electrochemical photo and solar cells. Principles and some experi-

ments, /. Electroanal. Chem., 58, 263 (1975).Green, M., Electrochemistry of the semiconductor-electrolyte interface, MAE, 2,

343 (1959).Hunter, R. J., The double layer in colloidal systems, CTE, 1, 397 (1980).Lyklema, J., The electrical double layer on oxides, Croatica Chem. Acta, 43, 249

(1971).Maredek, V., Z. Samec, and J. Koryta, Electrochemical phenomena at the interface

of two immiscible electrolyte solutions, Advances in Interfacial and ColloidScience, 29, 1 (1988).

244

Morrison, S. R., Electrochemistry of the Semiconductor and Oxidized MetalElectrodes, Plenum Press, New York, 1980.

Myamlin, V. A., and Yu. V. Pleskov, Electrochemistry of Semiconductors, PlenumPress, New York, 1967.

Pleskov, Yu. V., Electric double layer on semiconductor electrode, CTE, 1, 291(1980).

Samec, Z., The electrical double layer at the interface of two immiscible electrolytesolutions, Chem. Revs, 88, 617 (1988).

Spaarnay, M. J., see page 203.Vanysek, P., see page 191.Verwey, E. J. W., and J. Th. G. Overbeek, Theory of the Stability of Lyophobic

Colloids, Elsevier, Amsterdam, 1948.

Chapter 5

Processes in HeterogeneousElectrochemical Systems

In the case of electrodes with low overpotential the process of molecular hydrogenevolution is particularly more complicated than with electrodes showing highoverpotential where a single assumption of slow discharge step could successfullyelucidate all experimental results. It can be taken for sure, however, that in the caseof electrodes with low overpotential it is necessary to consider, as a slow step, theprocess of removal of molecular hydrogen from the electrode together with thedischarge process.

A. N. Frumkin, V. S. Bagotzky, and Z. A. Iofa, 1952

5.1 Basic Concepts and Definitions

This chapter will be concerned with the kinetics of charge transfer acrossan electrically charged interface and the transport and chemical processesaccompanying this phenomenon. Processes at membranes that often haveanalogous features will be considered in Chapter 6. The interface that ismost often studied is that between an electronically conductive phase(mostly a metal electrode) and an electrolyte, and thus these systems will bedealt with first.

A system consisting of two electrodes in an electrolyte medium is calledan electrolytic cell. It should be realized here that there is no basic differencebetween the concepts of a 'galvanic cell' and an 'electrolytic cell'. Incommon usage, the term 'galvanic cell' is understood to refer to a system inthe absence of current flow (it can be at equilibrium, as mentioned in theprevious chapter, or only in a steady state, as will be demonstrated inSection 5.8.4 on mixed potentials) or to a system that yields electric work toits surroundings; an 'electrolytic cell' is then a system receiving energy fromits surroundings to carry out the required chemical conversions. In fact,however, a given system sometimes can have both functions depending onthe electrical potential difference between the electrodes.

If an electrode reaction at a given electrode results in the transfer of apositive electric charge from the electrolyte to the electrode material or anegative charge from the electrode material to the electrolyte, then thecorresponding current is defined as cathodic (/c) and the process is termed a

245

246

cathodic reaction or reduction. In the opposite case, the current is termedanodic (/a) and the process is an anodic reaction or oxidation. The termscathode and anode have already been defined in Section 1.1.1. For example,in a Daniel cell (see Section 3.1.4), the cathode is the copper electrodewhen the cell carries out work; if the applied voltage is greater than theequilibrium value, current flows in the opposite direction and the cathodebecomes the zinc electrode at which zinc ions are reduced. According to theIUPAC convention, anodic current is considered to be positive and cathodiccurrent to be negative.

Generally, an electrode that is macroscopically characterized by a smoothsurface actually contains many steps and other microscopic irregularities.The real (physical) electrode surface Ar is thus mostly larger than thegeometric (macroscopic) surface Ag. The current density is usually definedas the current divided by the geometric surface area. The ratio of the realand geometric surface areas is termed the roughness factor, /R = AT/Ag. Thephysical and geometric surface areas are identical at mercury and otherliquid electrodes.

The flow of electric current through the electrolytic cell is connected withchemical, electrochemical and physical processes which, as a whole, aretermed the electrode process. The main electrochemical step in the electrodeprocess is the actual exchange of charged species between the electrode andthe electrolyte, which will be termed the electrode reaction (charge transferreaction). Substances participating directly in the charge transfer reactionare termed electroactive. These substances can be either soluble or insolublein the electrolyte or electrode material. Common basic types of electrodereactions are as follows:

1. Reduction processes where the electrode is the cathode:(a) Reduction of ions or complexes to a lower oxidation state, the

reduction of inorganic or organic molecules; the reduced formremains in solution.

(b) Deposition of ions or complexes on the electrode with formation of ametallic or gaseous phase or of an amalgam.

(c) Reduction of insoluble compounds or surface films with formation ofa metal phase or an amalgam or an insoluble phase with differentcomposition.

2. Oxidation processes in which the electrode is an anode:(a) Oxidation of ions, complexes or molecules to a soluble higher

oxidation state.(b) Oxidation of the electrode material with formation of soluble ions or

complexes.(c) Oxidation of the electrode material with formation of insoluble

anodic films or oxidation of insoluble films or other insolublesubstance to form insoluble substances with a higher oxidation state.

Of these electrode reactions, two—one oxidation and one reduction—

247

form a pair in a reversible charge transfer reaction (e.g. the deposition ofmetal ions with subsequent amalgam formation and ionization of theamalgam with formation of metal ions in solution).

A more fundamental classification considers the character of the chargetransfer between the electrode and the electroactive substance:

1. The transfer of electrons or holes between the electrode and theelectroactive substance present in solution.

2. Transfer of metal ions from the electrolyte into the electrode and viceversa.

3. Emission of electrons from the electrode into the solution with formationof solvated electrons and the subsequent reaction between the solvatedelectrons and the electron 'scavenger' in solution.

These processes have various characteristic properties when they occur atmetallic or semiconductor electrodes and if they occur between partners(electroactive substances or electrons or holes in the electrode) that are inthe ground or the excited state.

The basic condition for electron transfer in cathodic processes (reduction)to an electroactive substance is that this substance (Ox) be an electronacceptor. It must thus have an unoccupied energy level that can accept anelectron from the electrode. The corresponding donor energy level in theelectrode must have approximately the same energy as the unoccupied levelin the substance Ox.

On the other hand, in oxidation processes, the electroactive substanceRed must have the character of an electron donor. It must contain anoccupied level with energy corresponding to that of some unoccupied levelin the electrode. Oxidation occurs through transfer of electrons from theelectroactive substance to the electrode or through the transfer of holesfrom the electrode to the electroactive substance.

Thus, an electrode redox reaction occurs according to the scheme

Red + unoccupied level *± Ox + occupied level (5.1.1)

This situation is depicted in Fig. 5.1. The occupied level in the substanceRed2 has an energy corresponding to the unoccupied level in the electrode.Thus, oxidation can occur (either through the transfer of an electron e~ tothe electrode or of a hole h+ from the electrode). On the other hand, theunoccupied level in the substance Oxx has too high an energy, so that it doesnot correspond to any of the occupied levels in the electrode as all theselevels lie below the Fermi level £F, while the energy of the unoccupied levelof the substance Oxt is far above this level. Reduction can thus not occur.The situation is the opposite for the substances Red2 and Ox2.

As was demonstrated in Section 3.1.2, the energy of the Fermi level isidentical with the electrochemical potential of an electron in the metal. Achange in the inner potential of the electrode phase by A(p (attained bychanging the potential difference of an external voltage source by AE =

248

unoccupiedenergy levels

occupiedenergy levels

metal solution

Fig. 5.1 Transfer of an electron (or a hole) between theelectrode and the electroactive substance in solution.

(According to H. Gerischer)

i.e. by polarizing the electrode to the potential E + AE) can shift eF tohigher or lower energies. If £F is shifted towards the occupied level in Ox!(that is A(j> is negative), then reduction of Oxl is made easier and oxidationof Redi becomes more difficult (or even completely impossible if eF isshifted sufficiently above the energy of Redt). A decrease in eF (through apositive A0) towards the occupied level in Red2 has the opposite effect.Obviously, oxidation and reduction occur simultaneously in the vicinity ofthe equilibrium potential (where the Fermi levels in the metal and solutionare close to one another).

Conditions are somewhat more complicated at semiconductor electrodes.In contrast to metal electrodes, where the donor and acceptor levels areconcentrated in a single energy band, the Fermi level for semiconductorslies in the forbidden band (see Figs 2.3 and 3.2), where there are noelectron energy levels. Consequently, in the vicinity of the equilibriumpotential where the electrochemical potentials of the electrons in solutionand in the metal (i.e. the Fermi level) are identical, no charge transferoccurs (provided that there are no electronic surface states at the interface;see Section 4.5.1). Charge transfer requires a large change in A0, where theoccupied electronic level in Red attains the level of the conductivity band inthe semiconductor or the unoccupied level in Ox attains the level of thevalence band. It should be recalled that the electrons and holes participatein charge transfer in the vicinity of the semiconductor-electrolyte interface,

249

where their energy is different from that in the bulk of the semiconductor asa result of band 'bending' (see Sections 4.5.1 and 5.10.4).

Conditions at the interface between two immiscible electrolyte solutions(ITIES) are analogous to those at the electronic conductor-electrolyteinterface. The charge carriers passing across the interface are primarily ionsand they are transferred to a region where they have a lower electrochemi-cal potential. The transfer of electrons from the species Redj in one phaserequires that the second phase contain the species Ox2 with an unoccupiedacceptor level with similar or lower energy compared with the donor level inRedx. A change in the electrochemical potential of the ions or electrons canbe achieved through a change in the potential difference between the twophases.

The presence of a concentration gradient of electroactive substances atthe phase boundary between the electrode and the solution or between twoelectrolytes, the presence of an electric field and, finally, the effect ofmechanical forces all lead to transport processes, described in Chapter 2. Itshould be pointed out that investigations in electrochemical kinetics areusually carried out with solutions containing an excess of indifferentelectrolyte compared to the electroactive substance, to exclude the effect ofmigration. The rule for the equivalence between the material flux of thecharged species and the electric current (Eq. 2.3.14) is a general formula-tion of the Faraday law. In its classical form, this rule is connected with aspecial case, i.e. material flux through the electrode-solution interface. Inthis special case the Faraday law deals with equivalence between thematerial yield in an electrode reaction and the charge passed:

1. The amount of substance converted in the electrolysis is proportional tothe charge passed.

2. The amounts of various substances converted by a given chargecorrespond to the ratio of the chemical equivalents of these substances.

The electric current connected with chemical conversion is termed thefaradaic current in contrast to the non-faradaic current required to chargethe electrical double layer. The equation for the electrode reaction isformulated similarly to that for the half-cell reaction (Section 3.1.4); for thecathodic reaction, the electrons are placed on the left-hand side of theequation and, for the anodic reaction, on the right-hand side.

Consider the electrode reaction of metal deposition:

Mn+ + ne-^M (5.1.2)

If current / flows through the solution for time ty then the charge It ispassed, corresponding to a transfer of —It/F moles of electrons. If therelative atomic mass of the metal is M, then a total amount of

(5.1.3,

250

is discharged. This equation follows directly from Eq. (2.3.14) for a singlesubstance where zt = n and I=jA, where A is the area of the electrodesurface.

If no side reactions occur at the electrode that would participate in theoverall current flow, then the Faraday law can be used not only tomeasure the charge passed (i.e. in coulometres; see Section 5.5.4) but alsoto define the units of electric current and even to determine Avogadro'sconstant.

If the electrolyte components can react chemically, it often occurs that, inthe absence of current flow, they are in chemical equilibrium, while theirformation or consumption during the electrode process results in a chemicalreaction leading to renewal of equilibrium. Electroactive substances mostlyenter the charge transfer reaction when they approach the electrode to adistance roughly equal to that of the outer Helmholtz plane (Section 5.3.1).It is, however, sometimes necessary that they first be adsorbed. Similarly,adsorption of the products of the electrode reaction affects the electrodereaction and often retards it. Sometimes, the electroinactive components ofthe solution are also adsorbed, leading to a change in the structure of theelectrical double layer which makes the approach of the electroactivesubstances to the electrode easier or more difficult. Electroactive substancescan also be formed through surface reactions of the adsorbed substances.Crystallization processes can also play a role in processes connected with theformation of the solid phase, e.g. in the cathodic deposition of metals.

In electrode processes, the overall ('brutto') reaction must be distin-guished from the actual mechanism of the electrode process. For example, bya cathodic reaction at a number of metal electrodes, molecular hydrogen isformed, leading to the overall reaction

2H3O+ + 2e-^ H2 + 2H2O (5.1.4)

The electroreduction of aldehydes and ketones often involves the formationof dimeric hydroxy derivatives—pinacols:

2C6H5CHO + 2e 4- 2 H + ^ C6H5CHOH.CHOHC6H5 (5.1.5)

The deposition of cadmium in the form of an amalgam from a cadmiumcyanide solution involves the overall reaction

Cd(CN)42~ + 2e-» Cd + 4CN~ (5.1.6)

However, this formulation yields very little information on the actualcourse of the reaction, i.e. of which partial processes it consists. Thisset of partial processes is termed the mechanism of the electrode reaction.In the first case, the electrode reaction at some electrodes involves theformation of an adsorbed hydrogen atom, followed by the recombinationreaction:

H3O+ + e-+ H(ads) + H2O(5 1 7)

2H(ads)-*H2 K }

251

(this process can occur through other mechanisms; see Section 5.7.1). In thesecond case, the reduction

C6H5CHO + H+ -» C6H5C+HOH

C6H5C+HOH + e ^ C6H5CHOH (5.1.8)

2C6H5CHOH-^ C6H5CHOH.CHOHC6H5

occurs at the mercury electrode under suitable conditions (pH, potential).The deposition of cadmium from its cyanide complex at a mercury electrodeis analogous. A solution containing cyanide ions in a concentration of1 mol • dm"3 contains practically only the Cd(CN)4

2~ complex. This complexcannot participate directly in the electrode reaction and thus the Cd(CN)2

complex enters the electrode reaction in a given potential range; theequilibrium concentration of this complex in solution is small. The reactionscheme can then be written as

Cd(CN)42" Cd(CN)3~ + CN" «± Cd(CN)2 + 2CN~

Cd(CN)2 + 2 e ^ Cd(Hg) + 2CN~ (5.1.9)

It is obvious that the electrode process in general is more complicatedthan would appear from simple overall schemes of electrode processes. Itoften happens that the substances that are the initial components in theoverall reaction are not those that participate in the actual electrodereaction; the latter are rather different substances in chemical equilibriumwith the initial components.

The case where the overall reaction involves several electrode reactions iseven more general. An example is the reaction of quinone (Q) at a platinumelectrode. This process occurs at pH < 5 according to the scheme

Two chemical equilibria participate in this process and, in addition, stepwisecharge transfer reactions occur. The product of the first electrode reactionundergoes a further electrode reaction.

Electrochemical kinetics is part of general chemical kinetics and has asimilar purpose: to determine the mechanism of the electrode process andquantitatively describe its time dependence. Mostly the study involvesseveral stages. Firstly it is necessary to determine the reaction path, i.e. todetermine the mechanism of the actual electrode reaction (for more detail,see Section 5.3.1), and the partial steps forming the overall electrode

252

process. The next task involves finding the step that is the slowest and,therefore, controls the overall rate of the process, i.e. the current. Theother, faster steps are then not apparent as the appropriate equilibrium isestablished (e.g. in the chemical reaction or the electrode reaction) orbecause they are negligible (e.g. convective diffusion in connection with avery slow charge transfer reaction, where the concentration gradient formedis removed by stirring the solution).

For more complex electrode mechanisms, it is useful to designate theindividual steps by symbols: E for an electrode reaction and C for achemical reaction. Thus, the mechanisms of the process (5.1.7) are denotedas EC, of (5.1.9) as CE, of (5.1.8) as CEC and of (5.1.10) as CECE.

In electrochemical kinetics, the concept of the electrode potential isemployed in a more general sense, and designates the electrical potentialdifference between two identical metal leads, the first of which is connectedto the electrode under study (test, working or indicator electrode) and thesecond to the reference electrode which is in a currentless state. Electriccurrent flows, of course, between the test electrode and the third, auxiliary,electrode. The electric potential difference between these two electrodesincludes the ohmic potential difference as discussed in Section 5.5.2.

If current passes through an electrolytic cell, then the potential of each ofthe electrodes attains a value different from the equilibrium value that theelectrode should have in the same system in the absence of current flow.This phenomenon is termed electrode polarization. When a single electrodereaction occurs at a given current density at the electrode, then the degreeof polarization can be defined in terms of the overpotential. The overpoten-tial r\ is equal to the electrode potential E under the given conditions minusthe equilibrium electrode potential corresponding to the considered elec-trode reaction Ee:

rj = E-Ec (5.1.11)

However, the value of the equilibrium electrode potential is often not welldefined (e.g. when the electrode reaction produces an intermediate thatundergoes a subsequent chemical reaction yielding one or more finalproducts). Often, an equilibrium potential is not established at all, so thatthe calculated equilibrium values must often be used.

It should also be recalled that, for many electrodes in various solutions,the open circuit electrode potential, i.e. the electrode potential in theabsence of current flow, is not the thermodynamic equilibrium but rather amixed potential, discussed in Section 5.8.4.

Polarization is produced by the slow rate of at least one of the partialprocesses in the overall electrode process. If this rate-controlling step is atransport process, then concentration polarization is involved; if it is thecharge transfer reaction, then it is termed charge transfer polarization, etc.Electrode processes are often classified on this basis.

253

References

Albery, W. J., Electrode Kinetics, Clarendon Press, Oxford, 1975.Bamford, C. H., and R. G. Compton (Eds), Electrode Kinetics—Principles and

Methodology, Elsevier, Amsterdam, 1986.Compton, R. G. (Ed.), Electrode Kinetics, Elsevier, Amsterdam, 1988.Damaskin, B. B., and O. A. Petrii, Introduction into Electrochemical Kinetics (in

Russian), Vysshaya shkola, Moscow, 1975.Gerischer, H., Semiconductor electrode reactions, AE, 1, 31 (1961).Gerischer, H., Elektrodenreaktionen mit angeregten elektronischen Zustanden,

Ber. Bunsenges., 77, 771 (1973).Hush, N. S. (Ed.), Reactions of Molecules at Electrodes, Wiley-Interscience, New

York, 1970.Kinetics and Mechanisms of Electrode Processes, CTE, 7 (1983).Krishtalik, L. I., Charge transfer in Chemical and Electrochemical Processes, Plenum

Press, New York, 1986.Morrison, S. R., Electrochemistry at Semiconductor and Oxidized Metal Electrodes,

Plenum Press, New York, 1980.Myamlin, V. A., and Yu. V. Pleskov, Electrochemistry of Semiconductors, Plenum

Press, New York, 1967.Southampton Electrochemistry Group (R. Greef, R. Peat, L. M. Peter, D. Pletcher,

and J. Robinson), Instrumental Methods in Electrochemistry, Ellis Horwood,Chichester, 1985.

Vetter, K. J., Electrochemical Kinetics, Academic Press, New York, 1967.Vielstich, W., and W. Schmickler, Elektrochemie II. Kinetik elektrochemischer

Systeme, Steinkopf, Darmstadt, 1976.

5.2 Elementary Outline of Simple Electrode Reactions

5.2.1 Formal approach

The basic relationships of electrochemical kinetics are identical with thoseof chemical kinetics. Electrochemical kinetics involves an additional para-meter, the electrode potential, on which the rate of the electrode reactiondepends. The rate of the electrode process is proportional to the currentdensity at the studied electrode. As it is assumed that electrode reactionsare, in general, reversible, i.e. that both the anodic and the oppositecathodic processes occur simultaneously at a given electrode, the currentdensity depends on the rate of the oxidation (anodic) process, va, and of thereduction (cathodic) process, vC9 according to the relationship

j = nF(va-vc) (5.2.1)

where n is the charge number of the electrode reaction appearing in thegeneral equation

2 v/(r Red, <± 2 v,. oOx, + ne (5.2.2)

where Red, and Ox, are the reactants and v, r and v, o are the stoichiometriccoefficients.

254

The rates of the partial anodic and cathodic processes at a given electrodepotential can often be expressed by the relationships

(5.2.3)

vc=K n c,v' °

where ka is the potential-dependent rate constant of the anodic process, kc

is the potential-dependent rate constant of the cathodic process, c, is theconcentration of reactant i, and the order of the electrode reaction withrespect to the ith reactant, v/ta or v /c, is given by the relationships

v,a = —(5.2.4)

dlnuc

In more complicated reactions, the reaction orders v /a and v, c need notand often do not correspond to the stoichiometric coefficients vir and vio.In contrast to the latter, the reaction orders can often be fractional or evennegative. The concentration of a given reactant can sometimes appear in theexpressions for both the anodic and the cathodic reaction rates.

If it is known which of the reactions determine the rate of the overallcomplex electrode process, then the concept of the stoichiometric number ofthe electrode process v is often introduced. This number is equal to thenumber of identical partial reactions required to realize the overallelectrode process, as written in an equation of type (5.2.2).t If the rateconstant of this partial rate-determining reaction is /ca, then ka — k'Jv. Thus,for example, if the first of reactions (5.1.7) is the rate-determining step inthe overall electrode process (5.1.4) then the stoichiometric number has thevalue v = 2.

5.2.2 The phenomenological theory of the electrode reaction

As the generality of equations of type (5.2.3) should not be exaggerated(e.g. in the presence of strong adsorption, such as in electrocatalyticprocesses, they are no longer valid), the basic features of electrochemicalkinetics will be explained by using the simple electrode reaction

Red<^Ox + Aze (5.2.5)

t A different formulation would be: the stoichiometric number gives the number of identicalactivated complexes formed and destroyed in the completion of the overall reaction asformulated with charge number n.

255

The rate of the oxidation reaction (i.e. the amount of substance oxidizedper unit surface area per unit time) is given by the relationship (cf. Eq.5.2.3)

Va = kaCRed (5.2.6)

and the rate of the reduction reaction by the relationship

vc = kccOx (5.2.7)

The oxidized and reduced forms can be dissolved in the electrolyte butcan also be present as a solid phase (metal, insoluble compound). In thelatter case, the concentration of these electroactive substances is constantand is set by convention equal to unity. The reduced form can also bedissolved in the form of an amalgam in mercury representing the electrodematerial.

The rate of the electrode process—similar to other chemical reactions—depends on the rate constant characterizing the proportionality of the rateto the concentrations of the reacting substances. As the charge transferreaction is a heterogeneous process, these constants for first-order processesare mostly expressed in units of centimetres per second.

Similarly as for chemical reactions, the rate constants of the electrodereactions can be written in terms of the Arrhenius equation

(5.2.8)

(5.2.9)

where Pa and Pc are the pre-exponential factors which are independent of theelectrode potential, and A/?a and AHC are the activation enthalpies of theoxidation (anodic) and reduction (cathodic) electrode reactions, respec-tively. The wavy line indicates that these quantities depend on the electricpotential difference between the electrode and the solution, A0 = 0(m) —0(1). Since, according to Eq. (3.1.69), this quantity is identical, except for aconstant, with the electrode potential, A$ = E •+• constant, the activationenthalpy depends on the electrode potential. As demonstrated in a numberof experiments, the rate of the electrode reaction is approximately anexponential function of the electrode potential; the rate of the cathodic(reduction) electrode reaction increases towards negative electrode poten-tial values, while the rate of the anodic (oxidation) electrode reactionincreases with increasing (positive) electrode potential. This dependence ofthe rate of the electrode reaction on the electrode potential appears in asimple functional dependence of the activation enthalpy. Thus, in anapproximation, the activation enthalpy of the cathodic reaction is expressedas

AHC = A//° + ocnFE (5.2.10)

256

where or is a constant termed the charge transfer coefficient. In general,however, this quantity is also a function of the electrode potential (seeSection 5.3).

The functional dependence of the activation energy of the anodicelectrode reaction can be derived as follows. According to the definition ofthe rate of the electrode reaction, the partial current density

Kcd (5.2.11)

corresponds to the anodic electrode reaction and the current density

-jc = nFkccOx (5.2.12)

to the cathodic electrode reaction. The overall current density is given bythe relationship

) = / a +/c = nF(kacRed - kccOx) (5.2.13)

In the absence of current (/ = 0), the system is in equilibrium but theelectrode reactions nonetheless proceed. Thus, similar to chemical equi-libria, the electrode equilibria have a dynamic character. Under equilibriumconditions

(5.2.14)

Substitution of Eqs (5.2.8), (5.2.9) and (5.2.10) into this equation yields

^ P e x p l ~ R T — ) C o x (5215)

and, after rearrangement,

aE.^=.^+«Iln&+SI,n^ (5.2.16)nF nF nF Fa nF cRed

The equilibrium electrode potential is given by the Nernst equation (cf.3.2.17),

E E° + l ^ £ ° '+ l ^ (5.2.17)ln £ + lnnF aRed nF cRed

where EOr is the formal potential of the half-cell reaction (5.2.5) (cf. Section3.1.5). These equations satisfy Eqs (5.2.8) and (5.2.9) when

A#a = A/fa* - (1 - a)nFE (5.2.18)

Obviously 0 < a r < l . For the conditional (formal) electrode potential weobtain

^ - ^ - ^ g (5.2,9)P"

257

If the system is at equilibrium at the formal potential, cOx = Red (see Eq.5.2.14) and thus

ka = kc = k^ (5.2.20)

The quantity k^ is termed the conditional (formal) rate constant of theelectrode reaction. It follows from Eqs (5.2.8), (5.2.9), (5.2.10) and (5.2.18)that

Substitution for Paexp(-AH°JRT) and Pcexp(-AH°JRT) from theseequations into Eqs (5.2.8) and (5.2.9), considering Eqs (5.2.10) and(5.2.18), yields the equations for the dependence of the rate constant on theelectrode potential:

[»-^<E-£"»] (5.2.22)

A further substitution from these equations into Eq. (5.2.13) yields thefundamental equation of electrochemical kinetics:

(5.2.24)

In addition to the thermodynamic quantity EOr, the electrode reaction ischaracterized by two kinetic quantities: the charge transfer coefficient a andthe conditional rate constant fc"0". These quantities are often sufficient for acomplete description of an electrode reaction, assuming that they areconstant over the given potential range. Table 5.1 lists some examples of theconstant A:°. If the constant k^ is small, then the electrode reaction occursonly at potentials considerably removed from the standard potential. Atthese potential values practically only one of the pair of electrode reactionsproceeds which is the case of an irreversible or one-way electrode reaction.

As mentioned above, at equilibrium two opposing currents pass throughthe electrode, with absolute values termed the exchange current. Theexchange current density j0 can be expressed at an arbitrary value of theequilibrium potential as a function of the concentrations of the oxidized and

Table 5.1 Conditional electrode reaction rate constants k^ and charge transfer coefficients a. (From R. Tamamushi)

SystemElectrolyte

solution Electrode Method

Cu(NH3)22 ++2e^Cu(Hg)

Fe3 ++e+±Fe2

1 M HCIO41 M HCIO41 M H2SO4

1 M N H 3 ,

1 M NH4C1I M K C I

0 . 5 M H C 1 O 4

I M H C I1 M K N O 3

1 . 1 M H C 1 O 4

I M K O H

1 M HC1O4

1 M tartaricacid

1 M HC1O4

1 M HC1O4

0.014 M NaNO3

1.05MNaNO3

AgHg

C pasteHg

HgPtCPtHgPt

HgHg

HgHgHgHg

25252520

20252135

24 ± 220

2220

22202222

0.025 ± 0.0050.35 ±0.033.8 x 1 0 4

1.1 xl0~ 2

2.1 x KT4

9 x 10~6

1.2 x l O ' 4

6.6 X10-2

1.31.2 xHT 2

2.09 x 10-3

1.83.2 x l O ' 3

5 x 10-2

3 x 10-3

0.14 ±0.020.28—

—0.500.590.490.28—

0.50—

—0.52

0.25-0.300.25-0.30

ChronopotentiometryFaradaic impedanceCurrent-potential curveFaradaic impedance

Faradaic impedanceCurrent-potential curveCurrent-potential curveFaradaic rectificationFaradaic rectificationVoltammetry at rotating

disk electrodeFaradaic impedanceFaradaic impedance

Faradaic impedanceFaradaic impedancePolarographyPolarography

259

reduced forms of the electroactive substance. Equation (5.2.24) gives

[ ^ ] (5.2.25)

Substitution for E - EOr from the Nernst equation (3.2.14) or (5.2.17) yields

^ (5.2.26)

The charge transfer coefficient a can be found from the dependence of ;0 oncOx or cRed.

It is often useful to express the current density as a function of theoverpotential rj = E — Ee (cf. Eq. 5.1.11) and of the exchange currentdensity. On substituting for k^ from Eq. (5.2.26) and for E - EOr,

RTE - E0' = r, + — In (cOx/cRed) (5.2.27)

into Eq. (5.2.24) we obtain the equation

(see Fig. 5.2). The j — rj dependence is termed the polarization curve(voltammogram). If the overpotential is small (r}<RT/nF), then, onexpanding the exponential function and neglecting all the terms in the seriesexcept the first two, we have

^ (5-2.29)

The value of the exchange current determines the deviation of theelectrode potential from the equilibrium value during the current flow. Thegreater the deviation, the slower the electrode reaction. It follows from Eq.(5.2.29) that

The reciprocal value of this differential quotient,

_ RTP nFj0

(5.2.31)

has the dimensions of resistance; it is termed the polarization resistance atthe equilibrium potential. This resistance is referred to the unit electrodearea.

260

8 -

4 -

I1 °

- 4 -

-8

1 1

—

-

f

1 ' /-/ i

/0.75 /

I^^*--~~~ 0.5

-

-

1 1 10.08-0.08 -0.04 0 0.04

Overpotential tyV

Fig. 5.2 Dependence of the relative currentdensity j/j0 on the overpotential rj according toEq. (5.2.28). Various values of the chargetransfer coefficient a are indicated at eachcurve. Dashed curves indicate the partial cur-rent densities (Eqs 5.2.11 and 5.2.12 for a =

0.5). (According to K. Vetter)

Plotting the overpotential against the decadic logarithm of the absolutevalue of the current density yields the Tafel plot' (see Fig. 5.3). Bothbranches of the resultant curve approach the asymptotes for \ri\»RT/F.When this condition is fulfilled, either the first or second exponential termon the right-hand side of Eq. (5.2.28) can be neglected. The electrodereaction then becomes irreversible (cf. page 257) and the polarization curveis given by the Tafel equation

= a+b\og\j\ (5.3.32)

261

- 2 0 0 -

-100-

.2 0

s•o

100-

1

-

-

"1

\

1

\

1

N\

* • *

1

1

—^

//

i / i

/

/

1 1

1

-

1

N. -

1 "-6 -5 - 4

Log current density-3 -2

Fig. 5.3 Tafel diagrams for the cathodic and the anodic current density fora = 0.25 and y0 = 1(T4 A • cm"2

For an anodic process, the quantities a and b are given by the relationships

2.3RTa— —

b =

— a)nF

2.3RT

1 - oc)nF

log/o (5.2.33)

(5.2.34)

and for cathodic processes,

2.3 RT<*= =-log/0

b= -

ocnF

2.3RT(5.2.35)

ocnF

Very often, the value of the formal electrode potential E'o is not knownfor an irreversible electrode reaction. The overpotential rj cannot, therefore,

262

be determined and the Tafel equation is then used as an operationalformula in the form

E = a + blog\j\ (5.2.36)

It is often useful to replace this equation by Eq. (5.2.24) modified for anirreversible process (occurring at large overpotentials), which has thefollowing form for cathodic processes:

j = -nFk^ exp [ ^ }- jcO x

/ ocn¥E\= -nFkcony exp ^ - - ^ - J c O x (5.2.37)

The quantity /cconv = k°~ txp(anFEOt/RT) is the value of the rate constantof the electrode reaction at the potential of the standard reference electrodeand will be termed the conventional rate constant of the electrode reaction.It can be found, for example, by extrapolation of the dependence (5.2.36)to E = 0, as

, _ -Q')E=O (t. ,^conv ~ j7 yj.Z,.Jo)

The existence of the exchange current was clearly demonstrated by Pleskovand Miller on experiments with a solution containing metal ions labelledwith a radioactive isotope. The solution was in contact with a diluteunlabelled amalgam of the same metal. The electrode reaction (proceedingin absence of an external voltage imposed on the metal) involved exchangeof the labelled ions for unlabelled atoms in the amalgam. The rate ofenrichment of the amalgam by the labelled isotope and its dependence onthe concentrations of the solution and amalgam were completely inagreement with the relationship derived for the exchange current (5.2.26).The experiments were carried out under strong stirring, to suppress theeffect of diffusion on the rate of the electrode process.

An electrode reaction in which the oxidized form accepts more than oneelectron usually proceeds as a series of one-electron reaction steps. As willbe demonstrated below, if the formal potentials of these partial electrodereactions satisfy certain conditions, then the electrode reaction simulates thetransfer of several electrons in one step (Eq. 5.2.5) and obeys Eq. (5.2.24).An example is the two-electron reaction of substance Au converted tosubstance A3 by the transfer of two electrons, where the reaction occursthrough the unstable intermediate A2:

A, + e *=^± A2 (5.2.39)*a, l

A2 + e *=^± A3 (5.2.40)

263

The charge transfer reaction (5.2.39) is characterized by the formalelectrode potential £?', the conditional rate constant of the electrodereaction kf and the charge transfer coefficient ocly while the reaction(5.2.40) is characterized by the analogous quantities E2', kf and a2. If therate constants of the electrode reactions, which are functions of thepotential, are denoted as in Eqs (5.2.39) and (5.2.40) and the concentra-tions of substances Al5 A2 and A3 are cu c2 and c3, respectively, then

- - = kc,\C\ + kcac2 - kalc2 - kaac3 (5.2.41)r

The concentration of the unstable intermediate A2 is obtained from thesteady-state condition,

dc2

— = 0 = kcAcx - kcac2 - kaJc2 + kaac3 (5.2.42)

Substitution for c2 in Eq. (5.2.41) yields

J kcikc2= / c (5.2.43)

^r ^c,2 ' ^a, l ^c,2 ~» a , l

It will be assumed that in a given potential range

£a,l«£c,2 (5.2.44)

This condition expresses the fact that reaction (5.2.39) determines the rateof the overall process. On neglecting kal in the denominator of Eq.(5.2.43), substitution of

(5.2.45)

yields

264

Introduction of new constants

= kY exp I RT J(5.2.47)

converts Eq. (5.2.46) to Eq. (5.2.24) for n = 2. A similar result would beobtained if the rate of the overall reaction were controlled by the reaction(5.2.40). Thus, assuming the validity of Eqs. (5.2.42) and (5.2.44), thestepwise transfer of electrons can formally yield the same dependence of thecurrent on the potential as for direct transfer of several electrons in onestep.

This is a very common case in irreversible, multielectron electrodereactions. The rate-controlling process is often the first step in which, forexample, a single electron is accepted, while the other reactions connectedwith the acceptance of further electrons are very fast and the rate of thereverse (here oxidation) reaction is negligible. Equation (5.2.46) gives

The overall charge number of the electrode reaction is then n> while theexponential term in the rate constant of the electrode reaction has a formcorresponding to a one-electron reaction. If the value of EOf is not known,then the conventional rate constant of the electrode reaction is introduced,kcony = kf'exp(alFE%'/RT), so that Eq. (5.2.48) can be expressed in theform

j = - nFkcony exp ( - ~ " ) c i (5.2.49)

If the single-electron mechanism has not been demonstrated to be therate-controlling process by an independent method, then, in the publicationof the experimental results, it is preferable to replace the assumed quantityocx by the conventional value cm, provided that the charge number of theoverall reaction is known (e.g. in an overall two-electron reaction it ispreferable to replace 0^ = 0.5 by or = 0.25). If the independence of thecharge transfer coefficient on the potential has not been demonstrated forthe given potential range, then it is useful to determine it for the givenpotential from the relation for a cathodic electrode reaction (cf. Eq. 5.2.37):

2.3*rdlog| / |a^^F^E~ (5-250)

265

A similar relationship is valid for the charge transfer coefficient of theanodic electrode reaction, a,a.

The determination of the activation energies of electrode reactions isespecially important for the theory of electrode reactions and for study ofthe relationship between the structure of the reacting substances and theelectrode reaction rates.

Several descriptions of electrode reaction rates discussed on the precedingpages and the difficulty to standardize electrode potential scales with respectto different temperatures imply several definitions of activation energies ofelectrode reactions. The easiest way to determine this quantity, for example,for an irreversible cathodic process, employs Eqs (5.2.9), (5.2.10) and(5.2.12) at a constant electrode potential,

AH°c+anFE A//*( 5 Z 5 1 )

This apparently simple expression is actually only conventional; it must berealized that the electrode potential at various temperatures does not differby only an additive constant when it is referred to different standardreference electrodes, but also by the product of the change in thetemperature and the temperature coefficient of this additive constant (seeSection 3.1.4). Often, however, the actual value of the activation energy atconstant potential AH\ is not of interest, but rather its increase AAi/f,giving the change in the activation energy between two members, e.g. in ahomologous series of electroactive substances (see Section 5.9.1). Thisquantity does not depend on the electrode potential if the charge transfercoefficient a is constant for the electrode process in the studied group ofsubstances. In this case

_AAfl?AAHf( 5 2 5 2 )dT ) E RT2 RT2 ( 5 - 2 5 2 )

The standard activation energy of the electrode reaction Hi is defined as

Equation (5.2.21) gives

d In fc° _ AHl - (1 - a)nFEOf AH°C + cmF£0'57 " RT2 ~ RT2 (5.2.54)

Elimination of E0' from this equation yields

Hi = aAHl + (1 - a)AH()c (5.2.55)

The third type of activation energy is the activation energy at constantoverpotential (rj »RT/F), defined by the equation

266

Table 5.2 Activation energies of electrode reactions (From R. Tamamushi andfrom A. A. Vlcek)

System

Cu(NH3)22+ + 2e +± Cu(Hg)

E i ^ + e ^ E u "MnO4~ + e +± MnO4

2~Tl+ + e*±Tl(Hg)

Vni + e+±Vn

VIV + e«±Vm

Co(NH3)63+ + e *± Co(NH3)6

2+

Electrolytesolution

1 M NH3,1 M NH4C1

I M K C II M K O H0.33 M K2SO4

5 mM H2SO4

1 M HC1O4

0 . 1 M H 2 S O 4

0 . 1 4 M H C 1 O 4

1.26MNaClO4

Electrode

Hg

HgPtHg

HgHgHg

A/Z^kJ-moP1)

20.9

37.619.68.0

32.659.053.0"

a AH\; E = 0.333 V vs. SHE.

For an anodic process, this quantity is given by a relationship following fromEqs (5.2.28), (5.2.26), (5.2.53) and (5.2.55):

AHl = AHl - (1 - a)nFrj (5.2.57)

and for a cathodic process by the relationship

AHl = AHl + cmFri (5.2.58)

Table 5.2 lists examples of the activation energies of electrode reactions.

References

See references on page 253Delahay, P., Double Layer and Electrode Kinetics, John Wiley & Sons, New York,

1965.Parsons, R., Electrode reaction orders, charge transfer coefficients and rate

constants. Extension of definitions and recommendations for publication ofparameters. Pure AppL Chem., 52, 233 (1979).

Tamamushi, R., Kinetic Parameters of Electrode Reactions of Metallic Compounds,Butterworths, London, 1975.

Vlcek, A. A., Relation between electronic structure and polarographic behaviour ofinorganic depolarizers. VII. Determination of activation energy of electrodeprocesses. Coll. Czech. Chem. Comm., 24, 3538 (1959).

5.3 The Theory of Electron Transfer

5.3.1 The elementary step in electron transfer

This part is concerned with discussion of the reaction mechanism in anarrower sense, i.e. the 'microscopic' elementary step in the electrodereaction, connected with the transfer of a single electron between the

267

electrode and the oxidized or reduced form of the electroactive substance.The possibility of adsorption interaction between the electrode and theelectroactive substance is excluded.

The contemporary theory of chemical reactions attempts to describesimilar microscopic steps on the basis of construction of the potential energyhypersurface of the reacting system if the applicability of the Born-Oppenheimer approximation can be assumed. In this approximation, alsocalled the adiabatic approximation, the nuclear motion of the atoms(forming the molecules of the reactants and products) is supposedlyindependent of the motion of the electrons forming the electron shells ofthese atoms. The path of the reacting system across this hypersurface istermed the system trajectory. An advantageous system trajectory passingthrough a minimum in the valley of the energies of the reactants andproducts and through a saddle point on the hypersurface is the reaction pathof the system. Obviously, construction of the potential energy hypersurfaceof the system encounters numerous difficulties and is possible only in thesimplest case (reactions in the gaseous state). Thus, the discussion ofelementary processes is often limited to discussion of energy profiles, basedon a cross-section through the hypersurface along the estimated reactionpathway, where the resultant diagram, the reaction profile, describes onlythe dependence of the potential energy on the reaction coordinate. Thereaction coordinate indicates the change of a single space coordinate of thesystem during the reaction.

Attempts were made to quantitatively treat the elementary process inelectrode reactions since the 1920s by J. A. V. Butler (the transfer of ametal ion from the solution into a metal lattice) and by J. Horiuti and M.Polanyi (the reduction of the oxonium ion with formation of a hydrogenatom adsorbed on the electrode). In its initial form, the theory of theelementary process of electron transfer was presented by R. Gurney, J. B.E. Randies, and H. Gerischer. Fundamental work on electron transfer inpolar media, namely, in a homogeneous redox reaction as well as in theelementary step in the electrode reaction was made by R. A. Marcus (NobelPrize for Chemistry, 1992), R. R. Dogonadze, and V. G. Levich.

In polar media, the strong solvation interaction between the ions andsolvent dipoles, with an energy of the order of 10-102kJ • mol"1, must beconsidered for electron transfer between the electrode and the electroactiveion, or between two components of the redox system in solution. Thus,charge transfer is connected with a marked change in the solvation shell,and the activation energy of this transfer depends on the momentaryconfiguration of the solvation shell given by its thermal fluctuations.

A comparison of the transfer between two species in a vacuum and in thecondensed phase reveals marked differences. Figure 5.4A depicts the energylevels of an electron in two species, where level eA in species A is occupiedand level eB in species B is empty. If these levels had different energies, asindicated in the figure, then the non-radiant energy transfer would be

268

£rA(x0,i)

£B(x0li)

Fig. 5.4 Electron transfer in vacuo and in solution. (A) When the electrondonor and the proton acceptor have very different corresponding energylevels the electron transfer is impossible. (B) When the reaction partnersare present in a solution, then a change in their configuration can bringtheir corresponding energy levels close together (dashed line) so thatelectron transfer is possible. (C) An analogous situation is in the system of

an electrode and an electroactive particle in solution

connected with an energy change that would have to appear instantaneously(during 10"15-10"16s) in the kinetic energy of the reacting species.However, such a rapid change of velocity of the particles is impeded by theinertia of the particles, so that electron transfer between levels of differentenergies is highly improbable. It becomes probable only when both thelevels have approximately the same energy. This condition is termed theFranck-Condon principle (isoenergetic electron transfer). It is also valid forelectron transfer in solution (where it was applied first by W. F. Libby), butin this case the time-dependent interaction of the electron donor or acceptorwith the solvent dipoles plays a decisive role (Fig. 5.4B). Thus, by themomentary configuration of the solvation shell, which determines both eA

and eB> the system can approach the situation eA ~ eB (although in theground state £ A ^ £ B ) - This is also true of the system consisting of anelectron in the electrode and an electron in the electroactive particle insolution (Fig. 5.4C). Here, the energy level of the electron in the electrodeis given by the Fermi level (the participation of excited states will beconsidered later).

The development of the theory of the rate of electrode reactions (i.e.formulation of a dependence between the rate constants k.A and kc and thephysical parameters of the system) for the general case is a difficultquantum-mechanical problem, even when adsorption does not occur. Itwould be necessary to consider the vibrational spectrum of the solvationshell and its vicinity and quantum-mechanical interactions between thereacting particles and the electron at various energy levels in the electrode.

269

Configuration ol nuclei

Fig. 5.5 Energy profile of the transition of the reactant R to theproduct P. (According to R. A. Marcus)

The limitation of the problem to a single energy profile is depicted in Fig.5.5 as a symbolic dependence of the potential energy of the reacting systemon the reaction coordinate.

If, according to the Marcus theory, the system passes rapidly through theregion of intersection of the potential energy curves for the initial state (R)and for the final state (P), then there is insufficient time for the electrontransfer, so that the system mostly passes through the intercept into theupper part of the R curve; the probability of electron transfer is low,characterizing non-adiabatic electron transfer. When the system passesthrough the intercept region slowly (i.e. with a slow change in theconfiguration of the atomic nuclei), then electron transfer almost alwaysoccurs. This is characteristic of adiabatic electron transfer.

The slow process is termed reorganization of the system. According toMarcus, the following contributions are involved:

(a) Changes in the energy and entropy of the internal structure of thespecies as a result of changes in the bond lengths and angles

(b) Changes in the size of the reacting particle(c) Changes in the vibrational and orientational polarization of the vicinity

of the particle

Further considerations will deal only with the simple case where the slowprocess involves a change in the solvation shell of the reacting species, e.g.for the Fe3+(aq) and Fe2+(aq) ions.

The applicability of the assumption of an adiabatic process depends onthe vibration frequency of the slow system (i.e. of the solvation shell). The

270

criterion is the quantity kT/hco, where fi is Planck's constant divided by 2nand co is the angular frequency of vibration. At laboratory temperaturekT/hco =* 10~13. For the vibration of the solvent shell co«kT/h while forthe vibration of the electron shell co » kT/ft, which means that the electrontransfer can be considered as adiabatic. Under these conditions, a furthersimplification is possible as the whole problem can be treated classically, thevibrations of the solvation shell being discussed in terms of a set of classicalharmonic oscillations.

In order to describe properly the electron transfer between the electro-active ion Bz+ and the electrode

Bz+(l) + e(m)-*B(2-1)+(l) (5.3.1)

the following assumptions are necessary:

(a) The solvation shells of the ions Bz+ and B(2~1)+ vibrate harmonicallywith identical frequency co> so that the potential energies of bothsolvation shells can be expressed by identical mutually shiftedparabolas.

(b) Quantum-mechanical effects need not be considered in the regionaround the activated complex (in the vicinity of the intersection of thetwo parabolas).

(c) The kinetic energy of the system is identical in the initial and finalstates.

(d) Only electrons from the Fermi level in the electrode and from the redoxsystem in the solution in the ground state participate in the electrodereaction.

(e) The concentrations of the ions Bz+ and B(2~1)+ are equal to unity.

The profile of the potential energy Ep of the reacting system independence on the reaction coordinate x is shown in Fig. 5.6. The curvedenoted as R corresponds to the initial state of the system. The coordinatesof its minimum, i.e. the ground state of the system, are xQ(i), 0. The curvefor the final state P is shifted by a value corresponding to the differencebetween the final and initial energies of the electronic subsystem, AEC. Thecoordinate xo(f) corresponds to the minimum on the potential curve for thefinal state. The potential energy of the initial state of the system is given bythe equation

Ep:i = mco2[x-x0(i)]2 (5.3.2)

(where m is the effective mass of the vibrating system) and the energy of thefinal state by the equation

£P,f = \mco\x - jco(f)]2 + A£e (5.3.3)

At the intercept, the quantities Ep{ and Ep{ are equal to the activationenthalpy of the cathodic electrode reaction A//c. The corresponding

271

Dependence of the potential energy Ep ofm on the reaction coordinate x during el-

f AH i h h d i i i

Fig. 5.6 Dependence of ththe system on the reaction coordinate x during el-ectron transfer. AHC is the cathodic activation energy,JC* the reaction coordinate corresponding to the inter-section of R and P, AEC the change in the electronicenergy from the ground states of the system and ET

the reorganization energy

reaction coordinate is given by the equation

AHC = \mio\xx - jco(i)]2 = \mcD2[xx - xo(f)]2

Thus,X — 21-toW ~ •*ol.l/'J "*" ^el-^OVl/ ~

Substitution into Eq. (5.3.4) yields

(Et + AEe)2

AH =AEV

(5.3.4)

(5.3.5)

(5.3.6)

(5.3.7)

The difference between the electronic energies of the final and initial statesmust include the energy of ionization of the ion B(z~1)+ in vacuo (where itsionization potential is complemented by the entropy term TA5,), theinteraction energy of the ions B2+ and B(z~1)+ with the surroundings, i.e.the solvation Gibbs energies, and finally the energy of an electron at theFermi level in the electrode. These quantities can be expressed most simply

where Er is the reorganization energy of the system,

272

in terms of the electrochemical potentials of the appropriate species (cf. Eq.3.1.17 and 3.1.18):

AEe = ti&MV - ^ B - ( I ) " ^e(m) + F[0(m) - 0(1)] (5.3.8)

where pi0' denotes the 'formal' chemical potentials (for unit concentrations).It is assumed for simplicity that metal m is also the material of the

standard hydrogen electrode, for which (cf. Eq. 3.1.61):

>±M!n> (5.3.9)

Substitution of 0(1) into Eq. (5.3.8) together with the relationship 0(m) -0(m') = E yields

AEe = F(E-EOf) (5.3.10)

where EOr is the formal electrode potential of the redox electrode reactiondefined by Eq. (5.3.1) (formulated as a reversible reaction),

. o , f & - ' > + 0 ) ~ f & + 0 ) + p k O ) ~ W g ) ~ - i n^Ox/Red = ~ (5.3.11)t

Equation (5.3.6) then assumes the form

~ £ ° " / R e d ) ] 2 (5.3.12)4ET

The general relationship for the activation energy including both theelectrode reaction and the chemical redox reaction as derived by Marcus inthe form

&±^±^>! (5 .3.13)

where vvr and wp are the works required to bring the reactants together andto separate the products and AG° is the standard Gibbs energy change forthe reaction. For electrode reactions, the quantities wr and wp appear in theeffect of the electrical double layer on the rate of the electrode reaction (seeSection 5.3.2).

If the effect of the electrical double layer is neglected (e.g. at higherindifferent electrolyte concentrations), the rate constant of the cathodicreaction is approximately given by the equation

kc = rspa)K exp ( - J

where rs is the geometric mean of the radii of the oxidized and of the

273

reduced form of the electroactive species, r — (rslrs?2)1/2, p is the number of

electronic states per unit area in the conductivity band of the electrode, a) isthe vibrational frequency of the molecule in the solvation sheath and K isthe transmission coefficient, with a value of K ~ 1 for an adiabatic processand K «1 for a non-adiabatic process. As it is assumed that the electron istransferred to a species whose centre lies in the outer Helmholtz plane, thevolume concentration of the reactants is converted by the factor 2rs to thesurface concentration. The density of the electronic states available for thetransfer reaction is estimated as p/2. This is a considerable simplification; ina more exact approach, the quantity p should be multiplied by the Fermifunction (Eq. 3.1.10), ^(m) - F0(m) = fie(m) in Eq. (5.3.8) should bereplaced by the energy of an electron level e and the whole expressionshould be integrated with respect to e. However, the more precise solutionwould not lead to a great improvement because of the great many othersimplifications involved in the present treatment.

In the same way the equation for the rate constant of the anodic reactionis obtained:

ka = rspo)K exp [J ~ ^Ox/Red)] ]ARTEr J

(5.3.15)

The charge transfer coefficient defined by Eq. (5.2.50) is given withrespect to Eq. (5.2.12) as

(5.3.16)

and is in general not a constant quantity. This is apparent from Fig. 5.7,

1

0

-2

-4

-6

-8

-0.4 -0.2 0 0.2 0.4

Overpotential ^,V

Fig. 5.7 Dependence of the rate con-stant of the cathodic reaction Fe3+ +e-H>Fe2+ in 1 M HC1O4 on the over-potential. (According to J. Weber, Z.

Samec and V. Marecek)

274

depicting the dependence of the rate constant of the cathodic reduction ofFe3+(aq) ions on the potential.

Very often the reorganization energy is rather large (the barrier for theprocess is high), ET»F \E - Eox/Rcd\, and Eq. (5.3.14) attains a simplifiedform,

[ £ / 4 + ^ Er - £ — > ] (5.3.17)

which is identical with Eq. (5.2.23) for fc° = rspcojcexp {-EJ4RT), n = \and a = \. It should be recalled that the value a = \ is connected with theassumption of a single harmonic oscillator with the same frequency in theinitial and final states of the electrode reaction. In the general case whenthis assumption is not fulfilled, the charge transfer coefficient can assumevarious values between 0 and 1, as given in the description of thephenomenological theory (Section 5.2). At a sufficiently large value of thequantity A£e, the intercept between the curves R and P approaches theminimum on the curve of the final state P and, according to Eq. (5.3.16),the charge transfer coefficient approaches unity. In this situation, however,as shown by L. I. Krishtalik, the electrons from the region above the Fermilevel will participate in the process so that a further increase in AEe will notresult in a shift of the intercept in Fig. 5.6 along curve P to the right. On thecontrary, the activation energy will increase in the same way as A£e

increases, i.e. the charge transfer coefficient retains the value of unity. Thisbarrierless state has been observed for a number of electrode processes, e.g.for the hydrogen evolution reaction at mercury (see page 355).

The above-described theory, which has been extended for the transfer ofprotons from an oxonium ion to the electrode (see page 353) and some morecomplicated reactions was applied in only a limited number of cases tointerpretation of the experimental data; nonetheless, it still represents abasic contribution to the understanding of electrode reactions. Morefrequently, the empirical values n, A:° and a (Eq. 5.2.24) are the final resultof the investigation, and still more often only fcconv and an (cf. Eq. 5.2.49)or the corresponding constant of the Tafel equation (5.2.32) and thereaction order of the electrode reaction with respect to the electroactivesubstance (Eq. 5.2.4) are determined.

5.3.2 The effect of the electrical double-layer structure on the rate of theelectrode reaction

The electrode reaction occurs in the region where by the influence of theelectrode charge an electric field is formed, characterized by the distributionof the electric potential as a function of the distance from the electrodesurface (see Section 4.3). This electric field affects the concentrations of thereacting substances and also the activation energy of the electrode reaction,expressed by the quantities wr and wp in Eq. (5.3.13). This effect can be

275

explained in terms of the phenomenological theory (Section 5.2.2) by theprocedure proposed by A. N. Frumkin. In the electrode reaction thereacting particles approach the electrode to the smallest possible distancethey can reach without adsorption, i.e. to the outer Helmholtz plane. Theconcentrations of reacting particles at this distance from the electrode, c'Ox

and cRed, are given by Eq. (4.3.5) as

/ zOxFct>2\Cox = c O x e x p ^ - RT )

ZRedF<t>2

= c R e d expI" (zOx - n)F(j)2l

L RT J

The value of the electric potential affecting the activation enthalpy of theelectrode reaction is decreased by the difference in the electrical potentialbetween the outer Helmholtz plane and the bulk of the solution, 02> so thatthe activation energies of the electrode reactions are not given by Eqs(5.2.10) and (5.2.18), but rather by the equations

AHC = AH° + anF(E - 4>2)V S } (5.3.19)

A// a = AH°a - (1 - a)nF(E - <t>2)Under these conditions, Eq. (5.2.24) becomes

, ^ f [(1 —a)nF(E — d)2 —E°')l f (2: O x ~^)^02lj = zFk^xp ^ KRT *2 }- ] exp [ - > Ox ^2 '

RT

[ anF(E - $2- £°')-| /

-exp I RT J e x p I ~(5.3.20)

where ; ( 0 2 = 0) is the current density given by Eq. (5.2.24).

Figure 5.8 depicts the experimental Tafel diagram in log \j\ — Ecoordinates for the electrode reaction of the reduction of oxygen in0 . 1 M N H 4 F and 0 . 1 M N H 4 C 1 at a dropping mercury electrode (aftercorrection for the concentration overpotential, described in Section 5.4.3).As the reduction of oxygen to hydrogen peroxide is a two-electron process(see Eq. 5.7.7) with the rate controlled by the transfer of the first electron,where ka = 0, it can be described by a relationship formed by combinationof Eqs (5.2.48) and (5.3.20):

p f<E-^- ' ; 0 '>] . (5.3.21)

-160 -240 -320

Electrode potential, vs.NCE,mV

Fig. 5.8 Tafel diagram for the reaction O2 + 2e—»H2O2 on the mercury electrode for oxygenconcentration of 0.26 mmol • dm~3.• 0.1 M NH4F, dependence of current density /on E;C 0.1 M NH4F, dependence of y on E — 02;3 0.1 M NH4C1, dependence of; on E\O 0.1 M NH4CI, dependence of; on E — <p2-By means of the Frumkin correction, identical Tafel diagrams for different media are obtained.

(According to J. Kuta and J. Koryta)

-1.2

Electrode potential E,V

-1.4 -1.6

Fig. 5.9 Tafel diagram for the reaction S2O82 + 2e^> 2SO2

2 at a mercury electrode. Electrolyte: 1 mM Na2S2O8and varying concentrations of NaF: (1) 3, (2) 5, (3) 7, (4) 10, (5) 15, (6) 20mmol • dm 3. (According to A. N.

Frumkin and O. A. Petrii)

278

Correction for the effect of the potential difference in the diffuse layer iscarried out by plotting log |/| against E — (j)2 rather than against E. It can beseen from Fig. 5.8 that the corrected curves are identical for bothelectrolytes.

The effect of the electrical double layer on the reduction of polyvalentanions such as S2O8

2~ is especially conspicuous. Here it holds thataz -zOx<l. At potentials corresponding to a negatively charged electrode(E<Epzc), the potential of the outer Helmholtz plane is 02<O and itsabsolute value in dilute solutions increases with increasing negative poten-tials. This leads to a paradoxical situation: the rate of the cathodic reactiondecreases with increasing negative potential (see Fig. 5.9).

The Frumkin theory of the effect of the electrical double layer on the rateof the electrode reaction is a gross simplification. For example, theelectrode reaction does not occur only at the outer Helmholtz plane but alsoat a somewhat greater distance from the electrode surface. More detailedconsiderations indicate, however, that Eq. (5.3.20) can still be used todescribe the effect of the electrical double layer as a good approximation.

Various approaches to the theory of electrode reactions with participationof adsorbed react ants are beyond the scope of the present book. The effectof of the electrical double layer is negligible for chemisorption (see Section4.3.3 and 5.7.1).

For semiconductor electrodes and also for the interface between twoimmiscible electrolyte solutions (ITIES), the greatest part of the potentialdifference between the two phases is represented by the potentials of thediffuse electric layers in the two phases (see Eq. 4.5.18). The rate of thecharge transfer across the compact part of the double layer then dependsvery little on the overall potential difference. The potential dependence ofthe charge transfer rate is connected with the change in concentration of thetransferred species at the boundary resulting from the potentials in thediffuse layers (Eq. 4.3.5), which, of course, depend on the overall potentialdifference between the two phases. In the case of simple ion transfer acrossITIES, the process is very rapid being, in fact, a sort of diffusionaccompanied with a resolvation in the recipient phase.

References

All references from page 253Armstrong, F. A., H. A. O. Hill, and N. J. Walton, Reactions of electron-transfer

proteins at electrodes, Quart. Rev. Biophys., 18, 261 (1986).Delahay, P., see page 266.Dogonadze, R. R., Theory of molecular electrode kinetics, in Reactions of

Molecules at Electrodes, p. 135, see page 253.Dogonadze, R. R., L. I. Krishtalik, and V. G. Levich, Present state of the theory of

the elementary step of electrode reactions (in Russian), Zh. Vsesoyuz. Khim.Obsh., 16, 613 (1971).

279

Friedrich, B., Z. Havlas, Z. Herman, and R. Zahradnik, Theoretical studies ofreaction mechanisms in chemistry, Advances in Quantum Chemistry, Vol. 19 (Ed.P. O. Lovdin), Academic Press, Orlando, 1987.

Koryta, J., see page 191.Koryta, J., Ion transfer across water/organic solvent phase boundaries and analytical

applications, Part 1, 2, Selective El. Revs., 5, 131 (1983), 13, 133 (1991).Levich, V. G., Kinetics of reactions with charge transport, PChAT, IX B, 985

(1970).Marcus, R. A., Chemical and electrochemical electron-transfer theory, Ann. Rev.

Phys. Chem., 15, 155 (1964).Marcus, R. A., Electron, proton and related transfers, Faraday Discuss. Chem.

Soc, 74, 7 (1982).Marcus, R. A., and N. Sutin, Electron transfers in chemistry and biology, Biochim.

Biophys. Acta, 811, 265 (1985).Marecek, V., Z. Samec, and J. Koryta, see page 243.Newton, M. A., and N. Sutin, Electron transfer reactions in condensed phases,

Ann. Rev. Phys. Chem., 35, 437 (1984).Tsonskii, V. S., L. B. Korsunov, and L. I. Krishtalik, Electrochim. Acta, 36, 411

(1991).Vanysek, P., see page 191.

5.4 Transport in Electrode Processes

5.4.1 Material flux and the rate of electrode processes

Material changes occurring during electrolysis at electrodes lead to masstransport throughout the whole system. The mathematical treatment ofelectrode processes involves relationships between the concentrations, theirgradients and the current densities, corresponding to the rates of processesoccurring directly at the electrodes (primarily the rates of electrodereactions), as boundary conditions for the differential equation describingtransport processes in the electrolyte, formulated in Chapter 2. Sometimes(small currents, strong mixing), the concentration changes are so small thatthe effect of transport phenomena can be neglected.

According to Faraday's law, the current passing through the electrode isequivalent to the material flux of electroactive substances. The disap-pearance of electroactive substances in the electrode reaction is consideredas their transport through the electrode surface. Consequently, onlydiffusion and migration but not convection flux need be considered at theelectrode surface, as the electrode is impenetrable to the solutioncomponents.

Assume that both the initial substances and the products of the electrodereaction are soluble either in the solution or in the electrode. The systemwill be restricted to two substances whose electrode reaction is described byEq. (5.2.1). The solution will contain a sufficient concentration ofindifferent electrolyte so that migration can be neglected. The surface of theelectrode is identified with the reference plane, defined in Section 2.5.1. Inthis plane a definite amount of the oxidized component, corresponding tothe material flux /Ox and equivalent to the current density /, is formed or

280

disappears in unit time. The reduced form participates in the reverseprocess, connected with the material flux /Red. When both forms are presentin solution

j = nFJOx = -nFJRcd (5.4.1)

j=-nFDOx[—-\ =nFDKed[——\ (5.4.2)\ OX /x=o \ OX /x=o

In general, for a larger number of electroactive substances, the partialcurrent densities corresponding to the individual substances are additive.

It is necessary to clarify the meaning of the distance, x = 0, from theelectrode. We must remember that the concentration changes resulting fromtransport processes appear up to distances considerably larger than thedimensions of the electrical double layer (the space charge region, i.e. theregion of the diffuse electrical layer, is mostly spread out to a distance of atmost several tens of nanometres). Thus 'zero' distance from the electrodecorresponds to points lying just outside the diffuse electrical layer. At thisdistance, the concentrations of the electroactive substances are not yetaffected by the space charge. The influence of the surface charge is shown,however, in the rate constants of the electrode reactions which appear in theboundary conditions (see Section 5.3.2).

The current density corresponding to the electrode reaction (5.2.1) isdescribed by Eq. (5.2.13). Combination of Eqs (5.2.13) and (5.4.2) yields

dc^ = ^cCOx ~~ aCRed (5.4.3)x Z

kacRed (5.4.4)

If the rates of the electrode reactions are large and the system is fairly closeto equilibrium (the electrode potential is quite close to the reversibleelectrode potential), then the right-hand sides of Eqs (5.4.3) and (5.4.4)correspond to the difference between two large numbers whose absolutevalues are much larger than those of the left-hand sides. The left-hand sidecan then be set approximately equal to zero, A:ccOx — k3icKcd ~ 0, and in viewof Eqs (5.2.14) and (5.2.17),

r(£-£°>= exp —

F]

rk (5A5)cRed

which is a form of the Nernst equation. The electrode reaction thusproceeds approximately at equilibrium (this is often termed a 'reversible'electrode process).

When A —> o° or A —>0, then cRed—»0 or cOx—*0. A limiting current is thenformed on the polarization curve that is independent of the electrodepotential (see page 286). If the initial concentration, e.g. of the oxidized

281

component, equals c°, then the equations of the limiting diffusion currentdensity yd are obtained for various types of diffusion and convectivediffusion; these equations follow from Eqs (2.5.8), (2.7.12) and (2.7.17)after multiplication of the right-hand sides by — nF.

In the general case, the initial concentration of the oxidized componentequals Cox

ar*d that of the reduced component cRed. If the appropriatedifferential equations are used for transport of the two electroactive forms(see Eqs 2.5.3 and 2.7.16) with the corresponding diffusion coefficients, thenthe relationship between the concentrations of the oxidized and reducedforms at the surface of the electrode (for linear diffusion and simplifiedconvective diffusion to a growing sphere) is given in the form

(5.4.6)

For the sake of simplicity, it will further be assumed that cRed = 0.For the boundary conditions (5.4.5), i.e. the prescribed ratio of the

concentrations of the two forms, we may use the transformation

-A(DOx/DRed) 1/2cgl /21 + A(DOx/DRed)

Together with the boundary condition (5.4.5) and relationship (5.4.6), thisyields the partial differential equation (2.5.3) for linear diffusion and Eq.(2.7.16) for convective diffusion to a growing sphere, where D = DOx andc° = CoJ[l +k(DOx/DRed)

1/2]. As for linear diffusion, the limiting diffusioncurrent density is given by the Cottrell equation

jd=-nFclx(^} (5.4.8)

and, for convective diffusion to a growing sphere (cf. Eq. (2.7.17)), by theIlkovic equation of instantaneous current,

jd = -nFc0Ox(7DoJ3jtt)1/2 (5.4.9)

the current density at an arbitrary potential for a 'reversible' electrodeprocess is

/ = l + A(Poxd/£>Red)

1/2 ( 5 A 1 0 )

whence

O x ) I J v }

where / and /d are the currents corresponding to the current densities / and

282

For j = i/d, the half-wave potential Em is obtained:

Em_E + —ll (5.4.12)

As the ratio of the diffusion coefficients in Eq. (5.4.12) is mostly very closeto unity, Em~EOf.

For a 'slow' electrode reaction (boundary conditions 5.4.3 and 5.4.4)the substitutions, Eq. (5.4.6), kJkc = X, c = [1 + A(DOx/DRed)

1/2]cOx-A(DOx/Z)Red)1/2^ox and D = DOJ[1 + X(DOx/DKcd)

m] convert the bound-ary conditions (5.4.3) and (5.4.4) to the form

dc= 0, t > 0, D — = kccdx

(5.4.13)

while the partial differential equations (2.5.3) and (2.7.16) preserve theiroriginal form. The solution of this case is

anF(E-EOfYRT

-1 exp (Q2t) erfc (Qtm) (5.4.14)

where

anF(E - E0')RT

e x p(1 - a)nF(E - E0')

RT

(see Fig. 5.10)For large values of Qtm, this solution goes over to the diffusion-

0.8

0.6

0.4

0.2

0

1k i

-

i

2i

^ — - .

Q2.

• - 1

Qt 2

31

-

-

-

—

I0 1 2 3

Fig. 5.10 Dependence of Y = j/(nFcOxk^) exp [anF(E - E°')/RT] on Q2t andQtm. (According to W. Vielstich and H. Gerischer)

283

controlled case (Eq. 5.4.10). On the other hand, for small values of Qtm,the rate of the process is controlled by the rate of the electrode reactionalone,

j = -nFkcc°Ox (5.4.15)

The case of the prescribed material flux at the phase boundary, describedin Section 2.5.1, corresponds to the constant current density at theelectrode. The concentration of the oxidized form is given directly by Eq.(2.5.11), where K= -j/nF. The concentration of the reduced form at theelectrode surface can be calculated from Eq. (5.4.6). The expressions forthe concentration are then substituted into Eq. (5.2.24) or (5.4.5), yieldingthe equation for the dependence of the electrode potential on time (achronopotentiometric curve). For a reversible electrode process, it followsfrom the definition of the transition time r (Eq. 2.5.13) for identicaldiffusion coefficients of the oxidized and reduced forms that

Tl/2 _

(5.4.16)

(see Fig. 5.11). For an irreversible electrode reaction (A:a = 0),

QX

(5.4.17)

101

I"1i

+ 0.2

Fig. 5.11 Chronopotentiometric curves for a re-versible electrode reaction (Eq. 5.4.16) in E-t(upper curve) and E-tV2 (lower curve) coordinates.

T denotes the transition time

284

5.4.2 Analysis of polarization curves (voltammograms)

Electrode processes are often studied under steady-state conditions, forexample at a rotating disk electrode or at a ultramicroelectrode. Polarog-raphy with dropping electrode where average currents during the droptimeare often measured shows similar features as steady-state methods. Thedistribution of the concentrations of the oxidized and reduced forms at thesurface of the electrode under steady-state conditions is shown in Fig. 5.12.For the current density we have (cf. Eq. (2.7.13))

/ = = nFD0xdol[(c0x)x=o - c°Ox]

— nFKOx[(cOx)x==() — cO )

fdc*

(5.4.18)

( dc A\——— j = -nFZ) R e d 6 R e d [ (c R e d ) J C = 0 - cR e d]

= -nF/cR e d [ (cR e d ) J C = 0 ~ cLd] (5.4.19)

where <5Ox and <5Red are the thicknesses of the Nernst layer for the reducedand oxidized forms and KOX and icRed are the mass transfer coefficients.Again, the case cRed = 0 will be considered. The limiting cathodic current

Fig. 5.12 Steady-state concentration distribution of the oxidizedand of the reduced form of an oxidation-reduction system in the

neighbourhood of the electrode

285

density is

jd=-nFKOxc°Ox (5.4.20)

For the concentration at the surface of the electrode, Eq. (5.2.24)assumes the form

J - { [ f f >[ ^ < f f ^ ] } (5.4.21)

giving, after substitution for (cOx);t=0 and (cRed)A.=o from Eqs (5.4.18) and(5.4.19) and rearrangement,

This equation describes the cathodic current-potential curve (polarizationcurve or voltammogram) at steady state when the rate of the process issimultaneously controlled by the rate of the transport and of the electrodereaction. This equation leads to the following conclusions:

1. Limiting current. As the negative electrode potential increases, theright-hand side decreases to zero and the current density j approachesthe limiting value ;d. The limiting diffusion current does not dependeither on the potential of the electrode or on the rate of the electrodereaction as it is controlled by convective diffusion alone. According toEq. (5.4.20), it is directly proportional to the concentration of electroac-tive substance; this fact forms the basis of the application of limitingcurrents to analytical chemistry. The coefficient KOX is a function ofstirring; e.g. in the case of the rotating disk electrode (Eq. 2.7.13)

KOX = 0.62DZlv~l/6(om (5.4.23)

where the concentration is expressed in mol • cm"3, DOx in cm2 • s"1,y=rj/p in cm2 • s"1 with co in rad • s"1. The limiting current densityapparently increases with increasing stirring rate (for the rotating diskelectrode, it is proportional to the square root of the number ofrevolutions per second).

2. Reversible polarization curve (voltammogram). If the value of k^ is solarge that the first term on the right-hand side of Eq. (5.4.12) is muchlarger than the second term, even when j approaches yd, then

E = E°'+ — In + — iJ-^i (5.4.24)nF KOX nF j v

This is the equation of a reversible polarization curve. The anodicpolarization curve of the reduced form obeys an identical equation

286

i1

Curre

ntIU

8

6

4

2

—i 1 1 -j? r—

f />

i i ^ > ^ i i

f _---

0.4 0 -0.4

Electrode potential E,V

-0.8

Fig. 5.13 Steady-state voltammograms: (1) reversible voltam-mogram (Eq. 5.4.24); (2) quasireversible voltammogram (Eq.5.4.22) for /c° = 10~4cm • s"1; (3) irreversible voltammogram

° = l(T5cm • s"1. E°'(Eq. 5.4.26) for ' = -0AV, a = 0.3,4= 2 x 10~4 cm • s"1, ;d = 10~4 A • cm"

containing the anodic limiting current. This curve corresponds to the casethat equilibrium is established at the surface of the electrode between thereduced and oxidized forms according to the Nernst equation. Thedependence of the current density on the potential is not a function ofk°~. The reversible polarization curve has the form of a wave (see Fig.5.13, curve 1). The half-wave potential is given as

Em = E InnF KOX

(5.4.25)

This value is close to the formal electrode potential and independent ofthe convection velocity. The plot of log (;d —j)/j versus E is linear withthe slope nF/2.303RT.

3. Irreversible polarization curve (voltammogram). If the value of k^ is sosmall that the first term on the right-hand side of Eq. (5.4.22) is muchsmaller than the second term, even when / approaches ;d, then theequation assumes the form

KO

(5.4.26)

This irreversible polarization curve also has the shape of a wave (see Fig.5.13, curve 3) with the limiting current density yd. The half-wave

287

potential is

RT k^£1/2 = £° '+ — I n (5.4.27)

ant KOX

or, considering the definition of A:conv in Eq. (5.2.37),

RT ki\ ± aconv / c A /-*o \

El/2 = — - In (5.4.28)ocnF KOX

When increasing the rate of stirring the half-wave potential of areduction reaction is shifted to more negative values.

Obviously, in the case of a cathodic reaction, the process at morepositive potentials (at the 'foot' of the wave) is reversible while at morenegative potentials irreversible. Increasing rate of stirring makes theirreversibility more pronounced.

The constants characterizing the electrode reaction can be found fromthis type of polarization curve in the following manner. The quantity k^is determined directly from the half-wave potential value (Eq. 5.4.27) ifEOf is known and the mass transfer coefficient KOX is determined from thelimiting current density (Eq. 5.4.20). The charge transfer coefficient a isdetermined from the slope of the dependence of In [(/d —j)/j] on E.

At current densities well below the limiting value, j can be neglectedagainst yd in the numerator of the expression after the logarithm in Eq.(5.4.26); rearrangement then yields

RT RTE E°' + l C 1 * ^ l I/I < 5 A 2 9 )

This is the Tafel equation (5.2.32) or (5.2.36) for the rate of anirreversible electrode reaction in the absence of transport processes.Clearly, transport to and from the electrode has no effect on the rate ofthe overall process and on the current density. Under these conditions,the current density is termed the kinetic current density as it is controlledby the kinetics of the electrode process alone.

4. The quasireversible case. When Eq. (5.4.22) cannot be simplifiedas described under alternatives 2 and 3, then it should be considered thatthe first term on its right-hand side increases compared to the secondterm as the potential becomes more positive and decreases at morenegative potentials. As the velocity of convection increases, the secondterm increases compared to the first term.

For the sake of a practical analysis of the polarization curve theexponential dependence of the electrode reaction rate constant need not beassumed. Then the more general form of Eq. (5.4.22) can be written as

1 — " - ~ (I " exp [nF(E - Em)/RT]} (5.4.30)nFkcc°Ox j j d

288

This expression can be used to determine kc directly by extrapolation for/d"*00- F° r example in the case of the rotating disk this is achieved bymeans of the plot of; versus co~1/2.

In the case of an irreversible reaction the exponential term in Eq. (5.4.30)is negligible against unity.

5.4.3 Potential-sweep voltammetry

The transient method characterized by linearly changing potential withtime is called potential-sweep {potential-scan) voltammetry (cf. Section5.5.2). In this case the transport process is described by equations of lineardiffusion with the potential function

vt (5.4.31)

entering the potential-dependent parameters of boundary conditions(5.4.3)-(5.4.5). Here Ex is the initial potential (t = 0) and v is thepolarization rate dE/dt. The resultant I-E curve usually has a characteristicshape of a 'peak voltammogram' (see Fig. 5.14). At higher potentials thanthat of the peak the curve approaches the j ~ t~m dependence (Eq. (5.4.8)),while the part prior to the peak corresponds to a process dependent onelectrode potential (electrode reaction or Nernst's equilibrium) and

Potential E - E°v VFig. 5.14 Peak voltammogram (curve 1, j(t) — E dependence) and convoluted

voltammogram (curve 2, F(t) — E dependence) for the reversible case

no=—zj(u)

289

diffusion. The peak current density follows the Randles-Sevcik equation

;'P = 0A52(nF)3/2(RT)-m(Dv)mc°Ox (5.4.32)

The peak potential (corresponding to the peak current) for a cathodicreversible process is shifted by

AE = l.lRT/nF (5.4.33)

to more negative potentials in comparison with the reversible half-wavepotential (Eq. (5.4.12)).

A transformation of the peak voltammogram to the sigmoidal shapeshown in the preceding section, Fig. 5.13, is achieved by the convolutionanalysis method proposed by K. Oldham. The experimental functionj(t) =j[^(E — Ej)/v] is transformed by convolution integration

= Jt~m f [j(u)/(t - u)m]F(t) = Jt-l/2\ [j(u)/(t-u)l/2]du (5.4.34)

to a current function F(t) whose dependence on potential (Fig. 5.14, curve2) can be treated in the same way as the polarization curve in the precedingsection.

5.4.4 The concentration overpotential

The effect of transport phenomena on the overall electrode process canalso be expressed in terms of the concentration or transport overpotential.The original concentration of the oxidized form cOx decreases during thecathodic process through depletion in the vicinity of the electrode to thevalue (COX)A:=0

a nd, similarly, the concentration of the reduced form cRed

increases to the value (cRed)x=0. It follows from Eqs (5.4.18), (5.4.19) and(5.4.20) that

(r x _ r o (* J \\COx)x=0 —cOx\±~~ . J

Jd'c (5.4.35)

(cRed)x=0 = CRed( 1 ~ " j

where j d c is the cathodic limiting current density, given by Eq. (5.4.20), andj d a is the anodic limiting diffusion current density, given by the relationship

7d,a ~ "^^Red^Red ^J.4.J0J

The overpotential of an electrode process (Eq. (5.1.11)) is given in thecase of a simple first-order electrode reaction as

,, = E - ET = E - EOf - — l n ^ (5.4.37)

290

This equation can be expanded to give

] + ^ > V T ^ . ~ " X (5-4.38)

ij = f?r+?|c (5.4.39)The first two terms on the right-hand side of this equation express the

proper overpotential of the electrode reaction rjr (also called the activationoverpotential) while the last term, t]c, is the EMF of the concentration cellwithout transport, if the components of the redox system in one cellcompartment have concentrations (cOx)x=0 and (cKcd)xss0 and, in the othercompartment, CQX and c^ed. The overpotential given by this expressionincludes the excess work carried out as a result of concentration changes atthe electrode. This type of overpotential was called the concentrationoverpotential by Nernst. The expression for a concentration cell withouttransport can be used here under the assumption that a sufficiently highconcentration of the indifferent electrolyte suppresses migration.

By substitution of Eq. (5.4.37) into Eq. (5.4.21) and rearrangement weobtain

0i\l J C R e d

Substitution from Eqs (5.2.26) and (5.4.35) into Eq. (5.4.40) finally yieldsthe equation

This is the basic relationship of electrode kinetics including the concentra-tion overpotential. Equations (5.4.40) and (5.4.41) are valid for bothsteady-state and time-dependent currents.

References

Albery, W. J., see page 253.Bard, A. J., and L. R. Faulkner, Electrochemical Methods, Fundamentals and

Applications, John Wiley & Sons, Chichester, 1980.Delahay, P., New Instrumental Methods in Electrochemistry, Interscience, New

York, 1954.Levich, V. G., see page 81.Southampton Electrochemistry Group, see page 253.

5.5 Methods and Materials

The study of electrochemical kinetics includes measurement of variouselectrical data but, at the same time, it needs information on reaction

291

products and on the situation in the interphase. Thus, electrochemicalmethodology includes genuine electrochemical methods, analysis of theproducts of these processes, methods based on interaction of electromag-netic radiation, electrons or other charged species with the electrolysedsystem or with the electrode itself, etc.

Electrochemical methods have been described in a number of excellentmonographs (e.g. Bard and Faulkner, Albery, the Southampton Electro-chemistry Group, etc., see page 253). Thus, only the general characteristics ofthese methods will be given here. They mostly provide information on thefunctional relationships between the current density, electrode potential,time passed since the beginning of the electrolysis and sometimes also thetotal charge passed through the system. The use of these methods isconnected with two main problems:

1. Elimination of the ohmic electric potential difference in the electrolyticcell

2. Selection of a fixed or programmed electric variable

The electrode material and its pretreatment considerably influence thecourse of the electrode process. The importance of non-electrochemicalmethods is constantly increasing as a result of the development ofexperimental techniques.

5.5.1 The ohmic electrical potential difference

The origin of the ohmic potential difference was described in Section2.5.2. The ohmic potential gradient is given by the ratio of the local currentdensity and the conductivity (see Eq. 2.5.28). If an external electricalpotential difference AV is imposed on the system, so that the current / flowsthrough it, then the electrical potential difference between the electrodeswill be

AV = Ex - E2 + AVohm = Ex - E2 + IRt (5.5.1)

where Ex and E2 are the electrode potentials, &Vohm = IRt is the ohmicelectrical potential difference and /?, is the resistance of the electrolytic cell,measured between the surfaces of the two electrodes. This approachinvolves the average values of these quantities; the electrode potential andohmic electrical potential difference values can exhibit local variations inactual systems.

In the case of a cylindrical electrolytic cell with electrodes forming basesof the cylinder the ohmic potential difference is

AVohm = Ipl/S (5.5.2)

where / is the current, p the resistivity of the electrolyte, / the height and Sthe cross-section of the cylinder. Electrolytic cells with more diversifiedforms require more complicated approaches. For a centrosymmetrical

292

system, we have

dR = pdr/4jir2 (5.5.3)

and for the increment of the ohmic potential difference

dAVohm = Ipdr/4jtr2 (5.5.4)

The ohmic potential difference in an electrolytic cell consisting of a sphericaltest electrode, termed, for a small radius r0, ultramicroelectrode, in thecentre and another very distant concentrical counter-electrode is given bythe equation

0 (5.5.5)

Since the current isI = 4jirlj (5.5.6)

where j is the current density, the resulting expression for the ohmicpotential difference is

*Vohm = r0pj (5.5.7)

Obviously, the ohmic potential difference does not depend on the distanceof the counterelectrode (if, of course, this is sufficiently apart) being'situated' mainly in the neighbourhood of the ultramicroelectrode. Atconstant current density it is proportional to its radius. Thus, withdecreasing the radius of the electrode the ohmic potential decreases which isone of the main advantages of the ultramicroelectrode, as it makes possibleits use in media of rather low conductivity, as, for example, in lowpermittivity solvents and at very low temperatures. This property is notrestricted to spherical electroides but also other electrodes with a smallcharacteristic dimension like microdisk electrodes behave in the same way.

The kinetic investigation requires, as already stated in Section 5.1, page252, a three-electrode system in order to programme the magnitude of thepotential of the working electrode, which is of interest, or to record itschanges caused by flow of controlled current (the ultramicroelectrode is anexception where a two-electrode system is sufficient).

Figure 5.15 shows a simple arrangement of the three-electrode system.The reference electrode R is usually connected to the electrolytic cell througha liquid bridge, terminated by a Luggin capillary which is generally filledwith the solution being investigated. The tip of the Luggin capillary isplaced as close as possible to the surface of the electrode, A, whosepotential is to be measured (the test, working or indicator electrode). In thesimple case of measurement of stationary voltammograms, the electrodepotential is measured with a voltmeter with a high resistance P. Thecurrent passing through the system is controlled by the potential source Cand the resistance P. If this resistance is much higher than the resistance of

293

Fig. 5.15 Basic circuit for theelectrode potential measurementduring current flow: A is working(indicator, test), B, auxiliary andR, reference electrode connectedby means of the Luggin capillary(arrow) and P, potentiometer.The constant current source con-sists of a battery C and a variable

high-resistance resistor R'

the electrolytic cell, then the current is given by the ratio of the voltage ofsource C and resistance Rf.

5.5.2 Transition and steady -state methods

The classification of methods for studying electrode kinetics is based onthe criterion of whether the electrical potential or the current density iscontrolled. The other variable, which is then a function of time, isdetermined by the electrode process. Obviously, for a steady-state process,these two quantities are interdependent and further classification is un-necessary. Techniques employing a small periodic perturbation of thesystem by current or potential oscillations with a small amplitude will beclassified separately.

The controlled-potential methods require special experimental instrumen-tation for elimination of the ohmic potential difference, called thepotentiostat (see Fig. 5.16). The programmed voltage between the indicatorelectrode and the tip of the Luggin capillary is taken from a suitable sourceand fed to the input of the potentiostat. The current from the potentiostatwith feedback control then flows between the working and auxiliaryelectrodes. There are two main types of potentiostat differing in theirapplications. High-power potentiostats control the potential of an electrodethrough which a strong current passes that changes only slowly with time. It

294

Fig. 5.16 Block diagram of a poten-tiostatic network: for A, B and R seeFig. 5.15; P is a potentiostat based onan operational amplifier, G, generatorof programmed voltage and Z,

recorder

is used in preparative electrolysis, in research on galvanic cells, etc. Fastpotentiostats are used in the kinetic study of electrode processes wherethere are great variations in the current density during short time intervalsreaching fractions of a millisecond. The main factor limiting the usefulnessof this type of potentiostat is charging of the electrical double layer throughthe ohmic resistance of the electrolytic cell at the beginning of the pulse.

The study of processes at ITIES and in membrane electrochemistryrequires elimination of two ohmic potential differences, achieved with afour-electrode potentiostat, 'voltage-clamp' (Fig. 5.17).

Various types of controlled-potential pulsing are shown in Fig. 5.18. Thesimplest case is the single-pulse potentiostatic method (Fig. 5.18A). Thecurrent-time (I-t) curves obtained by this method have already beendescribed in Section 5.4.1, Eq. (5.4.10) and (5.4.14).

The double-pulse potentiostatic method (Fig. 5.18C) is suitable forstudying the products or intermediates in electrode reactions, formed in theA pulse by means of the B pulse. For example, if an electroactive substanceis reduced in pulse A and if pulse B is sufficiently more positive than pulseA, then the product can be reoxidized. The shape of the I-t curve in pulseB can indicate, for example, the degree to which the unstable product of theelectrode reaction is changed in a subsequent chemical reaction.

Methods employing individual linear or triangular pulses {potential-sweep, triangular pulse and cyclic voltammetry, sometimes also called

295

Fig. 5.17 A four-electrode potentiostaticcircuit (voltage clamp). Rx and R2 are ref-erence electrodes with Luggin capillaries (ar-rows) attached as close as possible to themembrane or ITIES (dashed line), Bx and B2are auxiliary electrodes, Pt and P2 potentio-stats, G programmed voltage generator and

Z recorder

potentiodynamic methods; Fig. 5.18D, E and G) yield information on theI-E dependence. Cyclic voltammetry involves the study of the curveobtained after a greater number of pulses, when a steady voltammogram isobtained. A certain disadvantage of these popular methods is the fact thatthe processes occurring at a given potential can depend on those that occurat preceding potentials (i.e. on the history of the electrode). Various typesof voltammetry with triangular pulses are applied to study the reactionproducts because each peak observed on the I-t curve usually correspondsto a definite electrode process. This is especially useful in the study ofprocesses at solid electrodes, which are almost always complicated byadsorption (see Section 5.7) and where quantitative analysis of the givenkinetic problem is often very difficult.

The term, polarography, is most often used for voltammetry withdropping mercury electrode (DME—Fig. 5.19A), which also belongs topotentiostatic methods, as practically constant potential is applied to theelectrode during the drop growth. The corresponding average current isautomatically recorded as a function of the slowly increasing potential. Thismethod was discovered by J. Heyrovsky (Nobel Prize for Chemistry, 1959).

Because of low current densities, the ohmic potential drop can often beneglected and a three-electrode system is not necessary. The same electrodecan act as the auxiliary electrode and the reference electrode (sometimes a

296

time time

.2

1 TLtime time

1time time

ttime

Fig. 5.18 Potentiostatic methods: (A) single-pulse method, (B), (C)double-pulse methods (B for an electrocrystallization study and C for thestudy of products of electrolysis during the first pulse), (D) potential-sweepvoltammetry, (E) triangular pulse voltammetry, (F) a series of pulses forelectrode preparation, (G) cyclic voltammetry (the last pulse is recorded),(H) d.c. polarography (the electrode potential during the drop-time isconsidered constant; this fact is expressed by the step function oftime—actually the potential increases continuously), (I) a.c. polarography

and (J) pulse polarography

large-surface area calomel electrode or mercury sulphate electrode is used).The average current at the dropping mercury electrode ( / ) is given by therelationship

Adt (5.5.8)

where j is the instantaneous current density, tx is the drop-time and A is theinstantaneous surface area of the electrode (see Eq. 4.4.4). If theapproximate equation (5.4.9) is used, then the Ilkovic equation is obtained

297

time

e c°nsiderJ

298

instead of the current density. The quantities ; and jd must be replaced by(/) and </d), KOX and icRed by KOX and KRed (kOx/kRed = (DOJDKed)

V2) andk^ by the product k^(A), where (A) is the mean area of the electrode (incm2):

,2/3,2/3(A)=0.51m2l3t (5.5.11)

with m in g • s l and tx in s.The theory of voltammetry with rotating disk electrode (RDE) was

described in Sections 2.7.2 and 5.4.1.The properties of the voltammetric ultramicroelectrode {UME) were

discussed in Sections 2.5.1 and 5.5.1 (Fig. 5.19). The steady-state limitingdiffusion current to a spherical UME is

Id = 4jtnFDrQc0 (5.5.12)

while to a hemispherical UME placed on an insulating support

Id = 2jtnFDroco (5.5.13)

B

glass tube -

coaxial cable

earth

soldered joint •

- soldered joint

copper wire

epoxide resin

hole tor epoxideinjection

• Wollaston wire

Fig. 5.19 Electrodes used in voltammetry. A—dropping mercury electrode(DME). R denotes the reservoir filled with mercury and connected by a plastictube to the glass capillary at the tip of which the mercury drop is formed.B—ultramicroelectrode (UME). The actual electrode is the microdisk at the tip

of a Wollaston wire (a material often used for UME) sealed in the glass tube

299

For a disk UME situated in the plane of the insulating support

Id = 4nFDroco (5.5.14)

The non-steady state situation at UME represents a comparatively simpleproblem at a spherical electrode while it is very difficult to solve in the caseof a disk electrode.

When comparing these three voltammetric electrodes they show differentperformance with respect to the purpose of their use.

1. With a DME, clean surface is quite easy to achieve whereas with theother electrodes a special treatment of the surface is required in order toobtain reproducible results (page 307). Even in solutions containingsurfactants, the amount adsorbed at DME can be controlled (cf. Section364).

2. The mass transfer coefficient for both RDE and UME can be varied in alarge interval while in the case of DME it is much smaller and also thepossibility of changing it is rather restricted.

3. In contrast to DME, the charging of the electric double layer makes noproblem with RDE and UME.

4. The work with both DME and RDE requires the use of a base(supporting or indifferent) electrolyte, the concentration of which is atleast twenty times higher than that of the electroactive species. WithUME it is possible to work even in the absence of a base electrolyte. Theohmic potential difference represents no problem with UME while in thecase of both other electrodes it must be accounted for in not sufficientlyconductive media. The situation is particularly difficult with DME.Usually no potentiostat is needed for the work with UME.

5. The area of the electrode can easily be measured with DME and RDEwhile in the case of very small UME it is often difficult to determine.

6. The high overpotential of hydrogen is an advantage of DME while thisproperty can be exploited only with mercury-covered RDE and UME.On the other hand, the dissolution of mercury at rather low positivepotentials is a disadvantage of DME which is not shared by RDE andUME made of nobler metals than mercury.

In differential pulse polarography, the dropping mercury electrode ispolarized by a sequence of pulses as shown in Fig. 5.18J. The differencebetween the currents flowing during several milliseconds before and at theend of a pulse is attenuated and recorded as function of the appliedpolarizing voltage, the charging current being thus eliminated. The resultantdependence of the current on the potential has the form of a peak, whose'height' is proportional to the concentration. The high sensitivity of thismethod ensures it an important position in electroanalysis.

The rotating ring disk electrode can be used to study the products of theelectrode reaction (Fig. 5.20). The products of the electrode reaction at thedisk are carried by convective diffusion to the ring where they undergo a

300

ring

Teflon

Fig. 5.20 The ring disksystem. Sometimes thedisk and the ring aremade of different

metals

further electrode reaction. The polarization curve at the ring is measured asa function of the current at the disk.

The simplest of the methods employing controlled current density iselectrolysis at constant current density, in which the E-t dependence ismeasured (the galvanostatic or chronopotentiometric method). The in-strumentation for this method is much less involved than for controlled-potential methods. The basic experimental arrangement for galvanostaticmeasurements is shown in Fig. 5.15, where a recording voltmeter oroscilloscope replaces the potentiometer. The theory of the simplest applica-tions of this method to electrode processes was described in Section 5.4.1(see Eqs 5.4.16 and 5.4.17).

Similar to potentiostatic methods, the main problem in the galvanostaticmethod is connected with charging of the electrode. This is especially truefor high current densities where only part of the charge is consumed as thefaradaic current and the rest is employed for charging the double layer, sothat the main condition for this method is not fulfilled. This disadvantagecan partly be removed by first applying a short impulse of strong current tocharge the electrode double layer and then employing a weaker constantcurrent (the galvanostatic double-pulse method).

The potential-decay method can be included in this group. Either acurrent is passed through the electrode for a certain period of time or theelectrode is simply immersed in the solution and the dependence of theelectrode potential on time is recorded in the currentless state. At a givenelectrolyte composition, various cathodic and anodic processes (e.g. anodicdissolution of the electrode) can proceed at the electrode simultaneously.The sum of their partial currents plus the charging current is equal to zero.As concentration changes thus occur in the electrolyte, the rates of thepartial electrode reactions change along with the value of the electrodepotential. The electrode potential has the character of a mixed potential(see Section 5.8.4).

In the closely related coulostatic method based on injection of a chargefrom a small condenser into an electrode in equilibrium with a redoxsystem. The resulting time dependence of the electrode potential originatesfrom the discharging of the electrical double layer by electrode reactions

301

that were initially in equilibrium. The resultant E-t dependence is afunction of the differential capacity of the electrode and the parameters ofthe electrode reaction. The advantage of this method is that no externalcurrent passes through the system so that no complicated instrumentation isrequired for elimination of the ohmic drop.

5.5.3 Periodic methods

If the working electrode-reference electrode-auxiliary electrode systemis polarized in a potentiostatic circuit (see page 293) by alternating voltagewith angular frequency co and small amplitude AE (AE<5mV), E +AE sin cot, then an alternating current, / = A/ sin (cot + 0), is obtained,where 6 is the phase angle. The electrode (or, in fact, the whole electrolyticcell) thus acts as an impedance (called the faradaic or electrochemicalimpedance) Z = Z' - )Z" = RS- j/(<oCs), where j = V11!, and Rs and Cs arethe resistance and capacitance in series connection. For the phase angle 6we have

0 = arctan (Z"/Zf) (5.5.15)

In an analysis of an electrode process, it is useful to obtain the'impedance spectrum'—the dependence of the impedance on the frequencyin the complex plane, or the dependence of Z" on Z', and to analyse it byusing suitable equivalent circuits for the given electrode system andelectrode process. Figure 5.21 depicts four basic types of impedance spectraand the corresponding equivalent circuits for the capacity of the electricaldouble layer alone (A), for the capacity of the electrical double layer whenthe electrolytic cell has an ohmic resistance RB (B), for an electrode with adouble-layer capacity CD and simultaneous electrode reaction with polariza-tion resistance Rp(C) and for the same case as C where the ohmic resistanceof the cell RB is also included (D). It is obvious from the diagram that theimpedance for case A is

coCL

for case B,

Z = RB--±r (5.5.17)O)Cr>

for case C,

(5.5.18)

Z * E + l + a;2C2Dfl2p l + co2C2

uR2p ( 5 ' 5 1 9 )

1 + a)2ClR2P 1 + co2ClRl

and, finally, for case D,

302

B

z'

1 cue -qp- -AAH h

Rp Rp

Fig. 5.21 Basic types of impedance spectra and of corresponding equivalentcircuits (Eqs 5.5.16 to 5.5.19). (According to R. D. Armstrong et al)

The calculation becomes more difficult when the polarization resistanceRP is relatively small so that diffusion of the oxidized and reduced forms toand from the electrode becomes important. Solution of the partialdifferential equation for linear diffusion (2.5.3) with the boundary conditionD(dcRcd/dx) = -D(dOx/dx) = A/sin cot for a steady-state periodic processand a small deviation of the potential from equilibrium is

AE =RTAI

n2FW/2 sin (5-5-20)

showing that the potential is behind the current by the phase angle 6 = 45°.When the rate of the electrode reaction is measurable, being charac-

terized by a definite polarization resistance RP (Eq. 5.2.31), the electrodesystem can be characterized by the equivalent circuit shown in Fig. 5.22.

303

Fig. 5.22 Equivalent circuit ofan electrode with diffusing el-ectroactive substances. W is theWarburg impedance (Eq. 5.5.21)

Here W is the Warburg impedance corresponding to the diffusion process

RT ( 1 1 \W % W 9 7 ^ b r ^ + o n i / 2 ( l - j ) = <7c»-1/2(l-j). (5.5.21)

The impedance of the whole system is given as

1(5.5.22)

The resulting dependence of Z" on Z' (Nyquist diagram) is involved but forvalues of Rp that are not too small it has the form of a semicircle withdiameter Rp which continues as a straight line with a slope of unity at lowerfrequencies (higher values of Z' and Z"). Analysis of the impedancediagram then yields the polarization resistance (and thus also the exchangecurrent), the differential capacity of the electrode and the resistance of theelectrolyte.

The impedance can be measured in two ways. Figure 5.23 shows animpedance bridge adapted for measuring the electrode impedance in apotentiostatic circuit. This device yields results that can be evaluated up to afrequency of 30 kHz. It is also useful for measuring the differential capacityof the electrode (Section 4.4). A phase-sensitive detector provides betterresults and yields (mostly automatically) the current amplitude and thephase angle directly without compensation.

A modification of faradaic impedance measurement is a.c. polarography,where a small a.c. voltage is superimposed on the voltage polarizing thedropping mercury electrode (Fig. 5.181).

5.5.4 Coulometry

If all the constants in the expression for the limiting diffusion current areknown, including the diffusion coefficient, then the number of electronsconsumed in the reaction can be found from the limiting current data.However, this condition is often not fulfilled, or the limiting diffusion

304

Fig. 5.23 A bridge for electrode impedancemeasurement in potentiostatic arrangement. D isthe bridge balance detector and E the variabled.c. voltage source. For A, B and R see Fig. 5.15.

(According to R. D. Armstrong et al)

current cannot be measured at all. The coulometric method is then used todetermine the electrochemical equivalent of the electroactive substance. Inthis method, the change of the concentration of the electroactive substanceis measured in dependence on the current passed through the electrolyticcell. The amount of charge is determined as the time integral of the current.

The electrolysis efficiency must be 100 per cent, i.e. the charge passedmust be consumed only in the studied electrode reaction. There are fourpossible sources of error:

(a) Electrolysis of water. The range of useful potentials is determined bythe potentials of oxygen and hydrogen evolution.

(b) Dissolution of the electrode material. This occurs particularly incomplexing media and leads to an increase of the anodic or decrease ofthe cathodic current.

(c) Subsequent electrode reaction of the products of electrolysis.(d) Electrode reduction of dissolved atmospheric oxygen.

Potentiostatic coulometry is mostly employed in the investigation ofelectrode processes. In this case, the current has to be directly proportionalto the concentration of the electroactive substance:

/ = AKC = —dm/dtzF (5.5.23)

305

where c is the instantaneous concentration of the electroactive species,dm/dt is the amount of substance reduced per unit time, A is the area of theelectrode and K is a proportionality constant; / and c are functions of timebecause of the solution being exhausted by the electrode process. If thevolume of the solution is V, then c = m/V and a first-order differentialequation

dm/m = -A/(zFV) dt (5.5.24)

is obtained, the solution of which gives an exponential dependence ofcurrent on time. The plot of log/ versus t has the slope — KA/ZFV, givingthe value of z. A uniform concentration in the cell is maintained by stirring.

Investigation of intermediates of an electrode reaction and rapid deter-mination of the electrochemical equivalents may be achieved by means ofthin-layer electrolytic cell only about 10 urn thick, consisting of twoplatinum electrodes which are the opposing spindle faces of an ordinarymicrometer.

5.5.5 Electrode materials and surface treatment

Metals are the most important electrode materials. Because of the readilyrenewable surface of mercury electrodes, they have been for severaldecades and, to a certain degree, still remain the most popular material fortheoretical electrochemical research. The large-scale mercury electrode alsoplays a substantial role in technology (brine electrolysis) but the generaltendency to replace it wherever possible is due to the environmentalharmfulness of mercury.

At mercury electrodes the geometric and actual surface areas areidentical. There is no site on the electrode that is energetically preferable toany other site (except for the deposition of heterogeneous patches). Exceptat rather positive potentials, the mercury electrode is usually not coveredwith a film of insoluble oxides or other compounds.

A liquid gallium electrode has similar properties to the mercury electrode(at temperatures above 29°C), but it has a far greater tendency to formsurface oxides.

Mercury electrodes require far less maintenance than solid metal electr-odes. Especially for the dropping mercury electrode, a noticeable amount ofimpurities present in the solution at low concentrations (<10~5 mol • dm"3)cannot appreciably reach the surface of the electrode through diffusionduring the drop-time (see Section 5.7.2).

Solid metal electrodes with a crystalline structure are different. Thecrystal faces forming the surface of these electrodes are not ideal planes butalways contain steps (Fig. 5.24). Although equilibrium thermal rougheningcorresponds to temperatures relatively close to the melting point, steps are acommon phenomenon, even at room temperature. A kink {half-crystalposition—Fig. 5.24c) is formed at the point where one step ends and the

306

Fig. 5.24 Crystalline metal face: (a) ad-atom, (b) ad-atom cluster, (c) a kink(half-crystal position), (d) atom at the step, (e) atom in the kink, (f) vacancy,

(g) vacancy cluster. (According to E. Budevski)

next begins, which is of great importance in crystal growth. In addition tosteps, the crystal plane also contains individual metal atoms {ad-atoms,Fig. 5.24a) and ad-atom clusters (Fig. 5.24b). A. Kossel and I. Stranski havestated that the individual atoms in the crystal plane have different energiesdepending on their degree of contact with the other atoms of the crystal.The atoms inside the crystal have the lowest energy, as would be expected,followed by the atoms incorporated in the crystal plane and then by atomsin steps and in kinks and by ad-atoms.

In addition, in the great majority of cases metal crystals containirregularities in the crystal lattice, termed dislocations. The role of screwdislocations will be considered in Section 5.8. In this type of dislocation, theindividual atomic planes in the crystal are mutually translocated by a smallangle, so that a wedge-shaped step arises from the crystal surface (Fig.5.25). A single-crystal electrode without dislocations can be made by themethod described by E. Budevski et al. A silver single crystal is fused in aglass tube with a capillary opening. This electrode is used as the cathode inthe electrolysis of AgNO3 by direct current with a small superimposed a.c.component. The front face then grows as a perfect, dislocation-free face,filling the cross-section of the capillary, usually approximately 0.2 mm indiameter.

Fundamental studies of the electrolyte solution/solid electrode interface

307

Fig. 5.25 A screw dislocation

can only be made by using well-defined single crystal electrodes. Thisadditionally requires not only electrochemical methods of characterization,but also other methods supplying detailed structural information (LEED,STM, etc., see Section 5.5.6).

Solid metal electrodes are usually polished mechanically and are some-times etched with nitric acid or aqua regia. Purification of platinum groupmetal electrodes is effectively achieved also by means of high-frequencyplasma treatment. However, electrochemical preparation of the electrodeimmediately prior to the measurement is generally most effective. Thesimplest procedure is to polarize the electrode with a series of cyclicvoltammetric pulses in the potential range from the formation of the oxidelayer (or from the evolution of molecular oxygen) to the potential ofhydrogen evolution (Fig. 5.18F).

The most important electrode material is platinum. Its widespreadelectrochemical use follows from its relatively high chemical inertness andelectrocatalytic properties. The available potential window of platinum (thepotential region where there is comparatively small faradaic current flow) isthe interval from about 0 to +1.6 V versus SHE in acidic aqueouselectrolytes; the negligible hydrogen overpotential of platinum makes itideally suitable for the preparation of a standard hydrogen electrode (seeSection 3.2.1). Platinum electrode is in aqueous electrolytes quicklycovered, according to the polarization voltage, either with adsorbedhydrogen or surface oxide; only in a narrow potential interval (from 0.4 to0.8 V versus SHE, the so-called 'double layer region') the Pt surface mightbe considered almost free from adsorbed or chemisorbed species. Inpractice, however, anions or organic impurities are always adsorbed even inthis potential region. The Pt surface particularly interacts with double bondsin organic molecules. Chemisorption of electroactive olefinic molecules hasalso been employed for the preparation of chemically modified electrodes(see below).

308

Platinum electrodes are made usually from polycrystalline metal; thecrystal planes at the surface include both the (111) and (100) faces inapproximately equal proportions. The electrochemical properties of Pt(lll)and Pt(100) faces are not identical. (Generally, the physical properties ofindividual metal crystal faces, such as work function, catalytic activity, etc.,are different.)

Cyclic voltammetry studies of single-crystal platinum electrodes in acidicaqueous electrolytes showed that the two characteristic peaks of hydrogenadsorption/desorption on platinum (see Fig. 5.40) correspond in fact toreactions at two different crystal faces: the peak at lower potential toPt(lOO) and the other one to Pt(lll).

The opposite case to Pt single crystal are electrodes whose surface areahas been increased by covering with platinum black. This so-calledplatinized platinum electrode (see page 173) shows up to 1000 x higherphysical surface area than the geometric surface area (the ratio of these twoquantities is called roughness factor, see page 246). An increase in thephysical surface area (roughness factor) of the electrode results in decreas-ing the relative coverage with surface-active impurities. The roughnessfactor of a single- crystal electrode is theoretically equal to one, but thisvalue can hardly be achieved in view of microscopic imperfections of thereal crystal face (see Fig. 5.24). A bright polycrystalline platinum electrodeshows a roughness factor 1.3-3 depending on the level of polishing.

The second most widely used noble metal for preparation of electrodes isgold. Similar to Pt, the gold electrode, contacted with aqueous electrolyte,is covered in a broad range of anodic potentials with an oxide film. On theother hand, the hydrogen adsorption/desorption peaks are absent on thecyclic voltammogram of a gold electrode in aqueous electrolytes, and theelectrocatalytic activity for most charge transfer reactions is considerablylower in comparison with that of platinum.

Semiconductors. In Sections 2.4.1, 4.5 and 5.10.4 basic physical andelectrochemical properties of semiconductors are discussed so thatthe present paragraph only deals with practically important electrodematerials. The most common semiconductors are Si, Ge, CdS, and GaAs.They can be doped to p- or n-state, and used as electrodes for variouselectrochemical and photoelectrochemical studies. Germanium has alsofound application as an infrared transparent electrode for the in situ infraredspectroelectrochemistry, where it is used either pure or coated with thintransparent films of Au or C (Section 5.5.6). The common disadvantage ofGe and other semiconductors mentioned is their relatively high chemicalreactivity, which causes the practical electrodes to be almost always coveredwith an oxide (hydrated oxide) film.

The surface reactivity is especially pronounced under illumination of thesemiconductor electrode with photons of an energy greater than the bandgap. The photogenerated minority carriers (electrons or holes) may react

309

with the electrode material to cause its photodecomposition, usually calledphotocorrosion (for electrochemical corrosion, see Section 5.8.4). It appearswith both p-type semiconductors (surface reduction), and n-type semicon-ductors (surface oxidation), but it is practically more important in the lattercase. The photocorrosion reactions can be demonstrated, e.g., on Si andCdS as follows:

(a) Cathodic photocorrosion of p-type materials

p-Si + 4e" + 2H 2 O^ SiH4 + 4OH"(5.5.25)

(b) Anodic photocorrosion of n-type materials

2n-Si + 4h+ + 2H 2 O^ SiO2 + 4H+

n-CdS + 2 h + ^ Cd2+ + S (5.5.26)

Here, the electrons are denoted e~ and the holes h+.The stability of n-Si, n-Ge, n-CdS, and n-GaAs photoanodes against

photocorrosion increases in the presence of a solution species that reactsrapidly enough with photogenerated holes to compete with the surfaceoxidation. An alternative way is to cover the semiconductor electrode withan electronically conducting polymer film, such as polyvinylferrocene,polypyrrole, polyaniline, Nafion/tetrathiafulvalene, etc. These films arecapable of efficiently capturing photogenerated holes and transport them tothe solution, where they undergo a useful redox reaction (photo-electrosynthesis). The removal of photogenerated holes suppresses not onlythe photocorrosion, but also electron-hole recombination within the illumi-nated semiconductor, which increases the quantum yield of the photo-electrochemical process. Protection of p-type semiconductors by the men-tioned conductive films is also possible; it enhances, for example, the rate ofH2 evolution at semiconductor photocathodes.

Various other semiconductor materials, such as CdSe, MoSe, WSe, andInP were also used in electrochemistry, mainly as n-type photoanodes.Stability against photoanodic corrosion is, naturally, much higher withsemiconducting oxides (TiO2, ZnO, SrTiO3, BaTiO3, WO3, etc.). For thisreason, they are the most important n-type semiconductors for photo-anodes. The semiconducting metal oxide electrodes are discussed in moredetail below.

Metal oxides. Noble metals are covered with a surface oxide film in abroad range of potentials. This is still more accentuated for common metals,and other materials of interest for electrode preparation, such as semicon-ductors and carbon. Since the electrochemical charge transfer reactionsmostly occur at the surface oxide rather than at the pure surface, the studyof electrical and electrochemical properties of oxides deserves specialattention.

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The electronic conductivity of metal oxides varies from values typical forinsulators up to those for semiconductors and metals. Simple classificationof solid electronic conductors is possible in terms of the band model, i.e.according to the relative positions of the Fermi level and theconduction/valence bands (see Section 2.4.1).

A transition behaviour between a semiconductor and a metal can also beobserved with metal oxides. This occurs when the Fermi level of thesemiconductor is shifted (usually by heavy doping) into the conduction orvalence band. This system is called degenerate semiconductor. If there existsa sufficient density of electronic states above and below thus shifted Fermienergy, the degenerate semiconductor can behave in contact with anelectrolyte as a quasimetal electrode (e.g., with the charge transfercoefficient a close to 0.5, cf. Section 5.2.8). The quasimetallic behaviour ofa degenerate semiconductor electrode is also conditioned by a sufficientcapacity of the electronic states near the Fermi energy. Assuming, forsimplicity, the double layer at the semiconductor/electrolyte interface as aparallel plate capacitor, whose capacity is independent of the potentialdifference across the electric double layer, the charge asc corresponding to apotential difference A|0 applied across the Helmholtz layer of thickness d is(cf. Section 4.5.1)

osc = As2<t>£80/d (5.5.27)

If the charge asc is fully accommodated by electronic states near the Fermienergy, no space charge is formed in the electrode phase, and any voltageapplied to the electrode appears exclusively across the Helmholtz layer, i.e.the system behaves like a metal.

Degeneracy can be introduced not only by heavy doping, but also by highdensity of surface states in a semiconductor electrode (pinning of the Fermilevel by surface states) or by polarizing a semiconductor electrode toextreme potentials, when the bands are bent into the Fermi level region.

Another possibility of the appearance of a quasimetallic behaviour ofmetal oxides occurs with thin oxide films on metals. If the film thickness issufficiently small (less than ca. 3 nm), the electrons can tunnel through theoxide film, and the charge transfer actually proceeds between the level ofsolution species and the Fermi level of the supporting metal (Fig. 4.12).

Although the band model explains well various electronic properties ofmetal oxides, there are also systems where it fails, presumably because ofneglecting electronic correlations within the solid. Therefore, J. B. Good-enough presented alternative criteria derived from the crystal structure,symmetry of orbitals and type of chemical bonding between metal andoxygen. This 'semiempirical' model elucidates and predicts electrical prop-erties of simple oxides and also of more complicated oxidic materials, suchas bronzes, spinels, perowskites, etc.

There is a number of essentially non-conducting metal oxides acting aspassive layers on electrodes; the best known example is A12O3. Metals that

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form insulating oxide films by anodic bias (e.g. Al, V, Hf, Nb, Ta, Mo) aretermed 'valve metals' (cf. Section 5.8.3).

Some insulating oxides become semiconducting by doping. This can beachieved either by inserting certain heteroatoms into the crystal lattice ofthe oxide, or more simply by its partial sub-stoichiometric reduction oroxidation, accompanied with a corresponding removal or addition of someoxygen anions from/into the crystal lattice. (Many metal oxides are,naturally, produced in these mixed-valence forms by common preparativetechniques.) For instance, an oxide with partly reduced metal cationsbehaves as a n-doped semiconductor; a typical example is TiO2.

Titanium dioxide is available as rutile (also in the form of sufficientlylarge single crystals) or anatase (only in polycrystalline form). Thirdmodification, brookite, has no significance for electrode preparation. Purerutile or anatase are practically insulators, with conductivities of the orderof 10~13S/cm. Doping to n-type semiconducting state is performed, forexample, by partial reduction with gaseous hydrogen, or simply by heatingat temperatures about 500°C. This is accompanied by a more or less deepcoloration of the originally white material, and by an increase of theelectronic conductivity by orders of magnitude. Semiconducting titaniumdioxide is one of the most important electrode materials in photo-electrochemistry (see Section 5.9.2).

Another example of a non-stoichiometric, n-semiconducting oxide isPbO2. This has been extensively studied as a positive electrode material forlead/acid batteries. Lead dioxide occurs in two crystal modifications,orthorhombic ar-PbO2, and more stable tetragonal j3-PbO2; both poly-morphs show high electronic conductivity of the order of 102 S/cm, and atypical mixed-valence Pb(II)/Pb(IV) composition with the O/Pb ratio in therange about 1.83-1.96. If the lead dioxide is prepared in an acidic aqueousmedium (as in a lead/acid battery), the oxygen deficiency is partlyintroduced by replacement of O by OH, i.e. a more correct formula of thismaterial is

Pb(IV)(1_;t/2)Pb(II);c/2H;cO2

Both modifications of PbO2 occur in the lead/acid battery in the approxim-ate ratio of a I(3 — 1/4. The standard potential of the ar-PbO2/PbSO4 redoxcouple in acidic medium is Eo= 1.697 V (for the /? modification by about10 mV lower). Although the lead dioxide is thermodynamically unstable incontact with water, its relatively high oxygen overvoltage is responsible for areasonable kinetic stability. A PbO2 layer supported by Pb is suitable foranodic oxidations of various inorganic and organic substances, since it isstable against further oxidation and its ohmic resistance is negligible.

Another class of conducting oxides are degenerate semiconductors,obtained by heavy doping with suitable foreign atoms. Two oxides, n-SnO2

(doped with Sb, F, In) and n-In2O3 (doped with Sn), are of particularinterest. These are commercially available in the form of thin optically

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transparent films on glass (Nesa, Nesatron, Intrex-K, etc.). Typical layerthickness is several hundreds of nm, and resistivities of up to 5-20Q/square. (Note: the unit 'Q/square' follows from the fact that theresistance of a uniform film on square support is independent of the size ofthe square, if we measure the resistance in direction parallel to the square.)

The electrochemical properties of conducting SnO2 and related materialshave been studied especially by T. Kuwana et al. Optically transparent SnO2

electrodes have found interesting applications in spectroelectrochemistry(see Section 5.5.6). Similar to PbO2, these electrodes exhibit relatively highoxygen overpotential and a good anodic stability, especially in acidicmedium. Their thermal stability is also satisfactory, i.e. SnO2 can be utilizedin electrolytes such as eutectic melts. The available potential window is,nevertheless, limited at the cathodic side by irreversible reduction of theoxide to the metal.

High electrical conductivity is also attained in oxides with very narrow,partially filled conduction bands; the best known example is RuO2. Thismaterial has a conductivity of about 2-3 104S/cm at the room temperature,and metal-like variations with the temperature. Some authors considerRuO2 and similar oxides as true metallic conductors, but others describethem rather as n-type semiconductors.

RuO2 is an important electrode material for industrial anodic processes.Special attention is deserved by the so-called dimensionally stable anodes(DSA) invented by H. B. Beer in 1968. These are formed by a layer of amicrocrystalline mixture of TiO2 and RuO2 (crystallite size less thn 0.1 /im)on a titanium support (Fig. 5.26). This material is suitable as anode forchlorine and oxygen evolution at high current densities. For industrialchlorine production, it replaced the previously used graphite anodes. These

titaniumsubstrate

Fig. 5.26 Beer-De Nora dim-ensionally stable electrode with a

RuO2-TiO2 layer

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were rapidly 'burnt' by evolved O2 and Cl2, and thus the cathode-anodespacing gradually increased, which caused an unwanted increase of voltageduring prolonged electrolysis.

Carbon. The electronically conductive carbons are derived from the hexag-onal crystalline modification—graphite.

(Note: Electronic conductivity, and even superconductivity at ca. 10 K,was observed with a new carbon polymorph, buckminsterfullerene, synthes-ized in 1990. Electrochemical and photoelectrochemical properties of asemiconducting diamond electrode were studied by Pleskov.)

Graphite is a thermodynamically stable form of elemental carbon belowapproximately 2600 K at 102MPa; its crystallographic description is D ^ -P63/mmc, unit cell constants: a{) = 245.6 pm, co = 670.8 pm. It shows amarked anisotropy of electronic conductivity: in the c-direction around1 S/cm (increasing with temperature as with semiconductors), but in thea-direction around 105S/cm (decreasing with temperature as with metals).The latter value can further be increased to a 106 S/cm level by intercalationwith AsF5 or SbF5 (see the discussion of intercalation compounds below).At least one of the graphite intercalation compounds (C8K) is a supercon-ductor at low temperature.

Large single crystals of graphite are rare; therefore, graphite is employedin polycrystalline form (solid blocks, paste electrodes, suspensions, etc.).Nevertheless, some carbons prepared by chemical vapour deposition(pyrolytic carbons) match the properties of a graphite single crystal. Thesupported layer of pyroiytic graphite is actually polycrystalline, but theindividual crystallites show a high degree of preferred orientation withcarbon hexagons parallel to the surface of the substrate. This is especiallytrue with the so-called highly oriented pyrolytic graphite (HOPG), preparedfrom ordinary pyrolytic graphite by heat treatment under simultaneouscompressive stress (stress annealing) around 3500°C. A HOPG containswell-ordered crystallites, whose angular scatter is typically less than 1°.

The faces parallel to the plane of carbon hexagons (100), also called basalplanes, show chemical and electrochemical properties significantly differentfrom those of the perpendicular (edge) planes (110). The latter containunsaturated carbon atoms (dangling bonds), which are highly active, forexample, against oxidation with oxygen, both chemical and electrochemical.The structure of thus formed surface oxides on carbons is very complex, butit can be derived from the common oxygen functional groups known fromorganic chemistry: carboxyl, hydroxyl, and carbonyl. These occur on theoxidized carbon surface either individually or in various combinations:surface anhydride, ether, lactone, etc. (see Fig. 5.27). Surface carbonyls(quinones) are reversibly reducible; further anodic oxidation of graphite andcarbons occurs through intermediates such as mellithic acid to finally giveCO2.

The basal plane of graphite (or HOPG) is essentially free from surfaceoxygen functionalities. This is manifested by almost flat, featureless

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EDGE PLANE

HOOC OH

BASAL PLANE

Fig. 5.27 Oxygen functional groups on the edge plane ofgraphite

voltammetric curves in aqueous electrolytes, and relatively low chargingcurrents. Unlike platinum, the basal plane HOPG virtually does not adsorbanions and most organic substances. Some molecules, such as phtalo-cyanines or porphyrines, form, however, n-n complexes with graphite, andare thus strongly adsorbed, being oriented parallel to the basal plane.

The most popular carbon material for electrochemistry is probably theglass-like carbon. (This term is preferable to the common names 'glassycarbon' and 'vitreous carbon', which are actually trade marks and shouldnot be used as terms.) Glass-like carbon is prepared by pyrolysis of suitablepolymers, e.g. polyfurfurylalcohol or phenolic resins. If the heating pro-gramme is properly controlled, the gaseous by-products of pyrolysis evolvesufficiently slowly and the final product is a monolithic, hard carbon,showing very low porosity and permeability to liquids and gases.

Glass-like carbon is, in contrast to graphite, isotropic in electricalconductivity and other physico-chemical properties. This explains itsstructure, characterized by randomly oriented strips (lamellae) of pseudo-graphitic layers of carbon hexagons. The lamellae occasionally approach thegraphitic interlayer spacing (335 pm), forming extremely small graphiticregions with crystalline dimensions La, Lc of the order of 1-10 nm. Theseregions are interconnected by a non-graphitic carbon material as shown inFig. 5.28. A similar three-dimensional network of graphitic microcrystallitescan also be found in other forms of 'amorphous' carbon, and, according tothe forces needed for breaking the network crosslinks (orientation ofcrystallites), we can classify almost all the solid carbons as hard (non-graphitizable) and soft (graphitizable) ones. Most polycrystalline carbons,including glass-like carbon, show an increase of electrical conductivity withincreasing temperature; they can be regarded as intrinsic semiconductorswith a band-gap energy of several tens to hundreds meV.

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Typical strong confluence

Weak confluence

Fig. 5.28 Structure of glass-like carbon. (According to G. M. Jenkins and K.Kawamura)

Carbon electrodes are widely used in electrochemistry both in thelaboratory and on the industrial scale. The latter includes production ofaluminium, fluorine, and chlorine, organic electrosynthesis, electrochemicalpower sources, etc. Besides the use of graphite (carbons) as a virtually inertelectode material, the electrochemical intercalation deserves special atten-tion. This topic will be treated in the next paragraph.

In spite of the high effort focused on the carbon electrochemistry, verylittle is known about the electrochemical preparation of carbon itself. Thischallenging idea appeared in the early 1970s in connection with the cathodicreduction of poly(tetrafluoroethylene) (PTFE) and some other perfluorin-ated polymers. The standard potential of the hypothetical reduction ofPTFE to elemental carbon:

(CF2)n + 2e" + 2H+ -> Cn + 2HF (5.5.28)

estimated from the Gibbs free energies of formation (cf. Sections 3.1.4 and3.1.5) equals 1.00 V at 25°C. The reductive carbonization of PTFE is,therefore, thermodynamically highly favoured, and it indeed occurs at thecarbon/PTFE interface in aprotic liquid or solid electrolytes. The producedelemental carbon is termed electrochemical carbon. It is reversibly dopable(reducible) under insertion of supporting electrolyte cations into the carbonskeleton.

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It should be emphasized that the electrochemical carbonization proceeds,in contrast to all other common carbonization reactions (pyrolysis), alreadyat the room temperature. This fact elucidates various surprising physico-chemical properties of electrochemical carbon, such as extreme chemicalreactivity and adsorption capacity, time-dependent electronic conductivityand optical spectra, as well as its very peculiar structure which actuallymatches the structure of the starting fluorocarbon chain. The electrochemi-cal carbon is, therefore, obtained primarily in the form of linear polymericcarbon chains (polycumulene, polyyne), generally termed carbyne. This canbe schematically depicted by the reaction:

F F F F F" F" F~ F"_ l _ l _ l _ l _ ^~~ f"~(f~~(f~~(f~" * = O = C - C ~ C _ (5.5.29)

F F F F F" F" F" F~The product of the reaction (5.5.29) is, however, unstable against sub-sequent interchain crosslinking and insertion of supporting electrolytecations. The electrochemical carbonization of fluoropolymers was recentlyreviewed by L. Kavan.

Insertion (intercalation) compounds. Insertion compounds are defined asproducts of a reversible reaction of suitable crystalline host materials withguest molecules (ions). Guests are introduced into the host lattice, whosestructure is virtually intact except for a possible increase of some latticeconstants. This reaction is called topotactic. A special case of topotacticinsertion is reaction with host crystals possessing stacked layered structure.In this case, we speak about 'intercalation' (from the Latin verb intercalare,used originally for inserting an extra month, mensis intercalarius, into thecalendar).

The intercalated guest molecules have a tendency to fill up completely theparticular interlayer space in the host crystal, rather than to occupyrandomly distributed interstitial sites between the layers. A peculiarity ofintercalation is that the sequence of filled and empty interlayer spaces isvery often regular, i.e. the structure of intercalation compounds consists ofstacked guest layers interspaced between a defined number of the hostlayers. The number of host layers corresponds to the so-called stages ofintercalation compounds. Stage 1 denotes successively stacked host andguest layers, stage 2 denotes a system with two host layers between oneguest layer, etc. This is schematically depicted in Fig. 5.29.

Graphite crystal is the best known host material for intercalation reac-tions. The host layers are clearly defined by planes of carbon hexagons, alsocalled graphene layers. (The term 'graphene' has been derived by analogywith condensed aromatic hydrocarbons: naphthalene, anthracene, tetra-cene, perylene, etc.). In contrast to other host materials (e.g. inorganicsulphides), graphite also forms compounds with high stage numbers

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guest

\oooooo oooooo oooooooooooo oooooo ooooooo o o o o o r:/::;:::/::::::

h o s t STAGE 1 STAGE 2 STAGE 3

Fig. 5.29 Definition of stages of intercalationcompounds

(10,11,. . ,). Intercalation of some guest species, such as alkali metals, cansimply be performed via a chemical reaction of a gaseous reactant withgraphite.

Some other guest species (e.g. H2SO4, HC1O4 and other inorganic acids),however, do not react spontaneously with graphite, but the intercalation canbe induced by an auxiliary oxidizing or reducing agent. The redox reactionpromo ting intercalation can also be performed electrochemically; theadvantage of electrochemical intercalation is not only the absence of anyforeign chemical agent and the corresponding reaction by-products, but alsoa precise control of potential, charge and kinetics of the process.

The electrochemical intercalation into graphite leads in the most simplecase to binary compounds (graphite salts) according to the schematicequations:

Cn + M+ + e"-> M+C~ (5.5.30)

Cn + A--e^CA- (5.5.31)

In the first case, the graphite lattice is negatively charged and the electrolytecations, M+ (alkali metals, NR| , etc.) compensate this charge by beinginserted between the graphene layers. In the second case, anions, A"(CIOJ, AsF^, BF4", etc.) are analogously inserted to compensate thepositive charge of the graphite host.

Reactions (5.5.30) and (5.5.31) proceed prevailingly during intercalationfrom solid or polymer electrolytes (cf. Section 2.6) or melts. When usingcommon liquid electrolyte solutions, a co-insertion of solvent molecules(and/or intercalation of solvated ions) very often occurs. The usual productsof electrochemical intercalation are therefore ternary compounds of ageneral composition:

M+(Solv)>;C- or C(Solv)yA~

The electrochemical intercalation of HSO4 anions together with H2SO4

was described by Thiele in 1934. The composition of the product ofprolonged anodic oxidation of graphite in concentrated sulphuric acid is

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and it is the stage 1 intercalation compound. By a very rapid heating of thiscompound, the intercalate vaporizes and forces the graphene layers apart,like in an accordion. The product is called exfoliated graphite', it exhibitsvarious peculiar properties, for example, up to hundred times smallerdensity than graphite. It is used, for example, in the preparation of graphitefoils.

The electrochemical intercalation/insertion has not only a preparativesignificance, but appears equally useful for charge storage devices, such aselectrochemical power sources and capacitors. For this purpose, theco-insertion of solvent molecules is undesired, since it limits the accessiblespecific faradaic capacity.

The electrochemical intercalation/insertion is not a special property ofgraphite. It is apparent also with many other host/guest pairs, provided thatthe host lattice is a thermodynamically or kinetically stable system ofinterconnected vacant lattice sites for transport and location of guestspecies. Particularly useful are host lattices of inorganic oxides andsulphides with layer or chain-type structures. Figure 5.30 presents anexample of the cathodic insertion of Li+ into the TiS2 host lattice, which ispractically important in lithium batteries.

The concept of electrochemical intercalation/insertion of guest ions intothe host material is further used in connection with redox processes inelectronically conductive polymers (polyacetylene, polypyrrole, etc., seebelow). The product of the electrochemical insertion reaction should also bean electrical conductor. The latter condition is sometimes by-passed, insystems where the non-conducting host material (e.g. fluorographite) isfinely mixed with a conductive binder. All the mentioned host materials(graphite, oxides, sulphides, polymers, fluorographite) are studied asprospective cathodic materials for Li batteries.

From this point of view, fluorographite, (CF^)^ with J C ^ I , deserves

Fig. 5.30 Insertion of Li+ between TiS2 hostlayers. (According to C. A. Vincent et aL)

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special attention. It is a covalent non-conductive compound whose structureis characterized by parallel planes of sp3 carbon atoms in trans-linkedperfluorinated cyclohexane chairs. The electrochemical reduction of fluor-ographite in a lithium cell was originally considered as a simple defluorina-tion with the formation of LiF and elemental carbon. Further studiesrevealed, however, that the primary process, controlling the discharge rateand the cell voltage, is an electrochemical Li+ intercalation between thelayers in the fluorographite lattice. A typical product of this reaction is aternary intercalation compound containing the co-inserted solvent mole-cules, Solv:

The electrochemical properties of fluorographite are also interesting inconnection with the electrolysis of melted KF-2HF, which is used forindustrial production of fluorine. Fluorine is here evolved at the carbonanode, which is spontaneously covered with a passivating layer of fluor-ographite; hence it causes an undesired energy loss during the electrolysis.

Chemically modified electrodes. Chemical and/or electrochemical surfacepretreatments are always needed in order to get reproducible data on solidelectrodes (see page 307); they are mostly focused on the removal of surfaceimpurities (oxides). The 'activation' of solid electrodes by (electro)chemicalpretreatment is, however, a complex and not well understood process. Forinstance, recent studies by scanning tunnelling microscopy demonstratedthat the common pretreatment of a platinum electrode by a series of cyclicvoltammetric pulses in the potential range of H2/O2 evolution (see page307) cannot simply be regarded as chemical 'purification' from adsorbedspecies, since it is accompanied with marked changes in surface topography.

A qualitatively new approach to the surface pretreatment of solidelectrodes is their chemical modification, which means a controlled attach-ment of suitable redox-active molecules to the electrode surface. Theanchored surface molecules act as charge mediators between the elctrodeand a substance in the electrolyte. A great effort in this respect wastriggered in 1975 when Miller et al. attached the optically active methylesterof phenylalanine by covalent bonding to a carbon electrode via the surfaceoxygen functionalities (cf. Fig. 5.27). Thus prepared, so-called 'chiralelectrode' showed stereospecific reduction of 4-acetylpyridine and ethylph-enylglyoxylate (but the product actually contained only a slight excess ofone enantiomer).

A large selection of chemical modifications employing the surface oxygengroups on carbon and other electrode materials (metals, semiconductors,oxides) has been presented. For instance, surfaces with a high concentrationof OH groups (typically SnO2 and other oxides) are modified byorganosilanes:

[S]—OH + Cl—SiR3-» [S]—O—SiR3 (5.5.32)

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where [S] stands for electrode surface, and the radical R might bear variouselectroactive species, such as ferrocene, viologenes, porphyrins, etc. Sur-faces with OH groups can also be modified via l,3,5-trichloro-2,4,6-triazineas a coupling agent. This further reacts with alcohols, amines, Grignardreagents, etc. Another method employes the surface carboxylic groups oncarbon electrodes, usually activated by conversion to acylchloride withSOC12, followed by esterification or amidization in order to anchor thedesired electroactive species.

In an ideal case the electroactive mediator is attached in a monolayercoverage to a flat surface. The immobilized redox couple shows asignificantly different electrochemical behaviour in comparison with thattransported to the electrode by diffusion from the electrolyte. For instance,the reversible charge transfer reaction of an immobilized mediator ischaracterized by a symmetrical cyclic voltammogram ( £ p c - Epa = 0; ypa =~7pc= l/pl) depicted in Fig. 5.31. The peak current density, /p , is directlyproportional to the potential sweep rate, v\

jp = n2F2Tv/ART (5.5.33)

where T is the surface coverage of electroactive species, which might simplybe determined from the voltammetric charge per unit electrode area (areaof the peak, see Fig. 5.31):

T=Q/nF (5.5.34)

or, alternatively, by non-electrochemical techniques of surface analysis(Section 5.5.6).

Besides the charge transfer reactions of the immobilized mediator, the

potential

- -y

Fig. 5.31 Cyclic voltammogram ofa chemically modified electrodewith a monolayer of a reversiblemediator. The shaded area cor-

responds to the charge Q

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mediated electron transfer to a solution species has to be considered. Apractically important situation occurs when the mediated electron transfer isfaster than the charge transfer reaction of the same solution species on abare electrode. If the charge transfer reaction at the electrode surface(which regenerates the original form of the mediator) is also faster, wespeak about an electron-transfer-mediated electrocatalysis.

Since 1975, hundreds of papers dealing with chemically modified elec-trodes have appeared in the literature. The initial effort in this topic wasreviewed by Murray in 1984. In subsequent years, the interest in surfacemodification gradually decreased. The reason might be that the ratherdemanding preparative techniques of chemical modification result, natur-ally, in a product possessing all the disadvantages of a two-dimensional(monolayer) system. The number of the redox-mediator turnovers, neededfor sufficiently high electrode currents, is, therefore, relatively very large.For instance, the mediated current of 1 A/cm2 would require (for a typicalmediator coverage F = 10~10mol/cm2), the turnover rate nt:

n^j/FT^lO's-1 (5.5.35)

which is beyond the practically available values. The surface modifiedelectrodes may also be rapidly deactivated by the mediator desorption ordecomposition as well as by adsorption of impurities from the electrolyte.

Therefore, recent interest has been focused prevailingly on electrodesmodified by a multilayer coverage, which can easily be achieved by usingpolymer films on electrodes. In this case, the mediated electron transfer tosolution species can proceed inside the whole film (which actually behavesas a system with a homogeneous catalyst), and the necessary turnover rate isrelatively lower than in a monolayer.

A discussion of the charge transfer reaction on the polymer-modifiedelectrode should consider not only the interaction of the mediator with theelectrode and a solution species (as with chemically modified electrodes),but also the transport processes across the film. Let us assume that asolution species S reacts with the mediator Red/Ox couple as depicted inFig. 5.32. Besides the simple charge transfer reaction with the mediator atthe interface film/solution, we have also to include diffusion of species S inthe polymer film (the diffusion coefficient DSp, which is usually much lowerthan in solution), and also charge propagation via immobilized redoxcentres in the film. This can formally be described by a diffusion coefficientDp which is dependent on the concentration of the redox sites and theirmutual distance (cf. Eq. (2.6.33).

The charge propagates in the film by electron hopping between thepolymer Red/Ox couples. This is controlled by the electrode potential onlyin a close proximity of the electrode; in more distant sites, the chargetransport is driven by a concentration gradient of reduced or oxidizedmediators. The observed faradaic current density, j F , is a superposition of

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Solution

Fig. 5.32 Scheme of the reduction of a solution speciesSox within the polymer film. Electron transfer is me-

diated by the immobilized redox active sites Red/Ox

these two contributions: diffusion of S and electron hopping in the film:

jF/nF = DSp(ds/dx)0 - Dp(dr/djc)0 (5.5.36)

(s and r are the concentrations of species S and Red/Ox sites, respectively).The solution of Eq. (5.5.36) may differ for various particular situations andboundary conditions, but we can phenomenologically describe the mediatedcharge transfer process in the polymer film by an effective heterogeneousrate constant kp and the concentration of S in solution, cs:

jF/nF = kpcs (5.5.37)

The redox active polymer films might bear the mediator group attachedeither covalently to the polymer backbone (polyvinylferrocene, Ru(II)complexes of polyvinylpyridine, etc.) or electrostatically within the ion-exchange polymer (e.g. in Nafion, cf. Section 2.6).

Polymer films on electrodes have been prepared by various techniques;the most simple one is dip coating, i.e. adsorption of a preformed polymerfrom its solution. Another method consists in spreading and evaporating thepolymer solution on the electrode. More uniform films are obtained by spincoating, which means evaporating on a rotating electrode. Polymer filmshave also been prepared by plasma polymerization, but the product ismostly ill-defined, owing to complicated side reactions introduced byhigh-energy particles in the plasma. Electrochemical polymerization de-serves special attention and will be treated in the next section.

Modification of electrodes by electroactive polymers has several practicalapplications. The mediated electron transfer to solution species can be usedin electrocatalysis (e.g. oxygen reduction) or electrochemical synthesis. Forelectroanalysis, preconcentration of analysed species in an ion-exchange filmmay remarkably increase the sensitivity (cf. Section 2.6.4). Various

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applications appeared also in corrosion protection, sensors, display devices,photoelectrochemistry, etc.

Electronically conducting polymers. The world-wide interest in electroni-cally conducting polymers has been triggered in 1977/8, when Heeger,MacDiarmid, Shirakawa, and others have studied the oxidation andreduction reactions of poly acetylene,

(-CH=CU-)n

This polymer occurs as trans- or a less stable ds-isomer. (The ds-isomercan be converted to trans- by heating.) Both isomers are intrinsic semicon-ductors with conductivities around 10~9 S/cm (cis) and 10~6 S/cm (trans).

Polyacetylene can partly be oxidized or reduced with considerableincrease in conductivity, up to 12 orders of magnitude. Oxidation orreduction of polyacetylene can be performed either chemically or electro-chemically; this process is often termed p- or n-doping. In both cases,the polymer backbone is virtually intact except for accumulating positive(p-doping) or negative (n-doping) charge. This charge is compensated byions of opposite sign, which are inserted into the polymer backbone eitherfrom a chemical dopant or from a supporting electrolyte.

The concept of p- and n-doping of polymers is similar to that of p- andn-doping of inorganic semiconductors, but it is, in fact, not equivalent. Forinstance, doping of silicon means that several atoms in the lattice of siliconare replaced by heteroatoms such as As or B (cf. Section 2.4, and Fig. 2.3).These sites are virtually electroneutral as long as they dissociate to form freeelectrons or holes and ionized heteroatoms (e.g. As+ or B~). On the otherhand, doping of polyacetylene and other conjugated polymers meansoxidation (p-doping) or reduction (n-doping) of the polymer skeletonassociated with insertion of a corresponding number of anions (cations) intothe polymer matrix. This can formally be described by Eqs (5.5.30) and(5.5.31) (where Cn stands for the polymer).

The individual macromolecular chains of conducting polymers agglomer-ate into more complicated structures, usually fibrous. The electronicconductivity of this system is a superposition of the conductivity of theindividual fibres (chains) and that due to electron hopping between thesedomains. The latter is usually much lower, i.e. it controls the totalconductivity of the system.

The electron hopping differs from the conventional solid-state conductionin that the charge transfer does not take place in the conduction band, butbetween localized states within the energy gap. This can be described byMott's model of variable range electron hopping. This model was originallyconsidered for non-crystalline inorganic semiconductors, such as amorphousgermanium or chalcogenide glasses, but it is readily applicable also toconducting polymers and other amorphous or pseudoamorphous solids withmicroheterogeneous distribution of conductive and insulating regions (a

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typical example is glass-like carbon, see Fig. 5.28, and other 'amorphouscarbons'). Mott's model follows from the preconception that the distance ofelectron hops varies with temperature and energy of the electron. Accord-ingly, the temperature dependence of conductivity of polymers (glass-likecarbon, etc.) can be expressed by the equation:

t- — A <=*YT> (cIh-Tn\ (*>> 1Q\iv ~~ -*i CAIJ I C / /v x I I J . J . JOI

where A is a constant and e is the activation energy of hopping. Theexponent n is usually 1/4 or 1/3 (it is the reciprocal of the effectivedimensionality of the space in which the charge transport occurs).

The charge transport in a conjugated chain and the interchain hopping isexplained in terms of conjugation defects (radical or ionic sites), calledsolitons and polarons. Several possible conjugation defects are dem-onstrated in Fig. 5.33 on the example of trans -polyacetylene.

A soliton is an electroneutral radical site with a spin equal to 1/2connected to the rest of the chain by somewhat longer bonds; the chargeand the bond lengths are, however, disturbed also in the close surroundingsof the defect, hence the soliton extends over about 15 (—CH—) units. If weremove or add an electron to the soliton, a zero spin and the correspondingcharge appears (charged soliton). Two neutral solitons can recombine andthe chain defect is cancelled. On the other hand, charged and neutralsolitons in neighbouring sites form a stable pair called polaron. Both thesolitons and polarons are mobile along the chain, but a soliton, unlike apolaron, moves virtually without overcoming energy barriers. If the number

trans- polyacetylene

soliton

charged soliton

polaron

bipolaron

Fig. 5.33 Conjugation defects in trans -polyacetylene

325

of polarons in the chain increases, they can interact basically in two ways:two polarons can either form two charged solitons or a new stable zero-spindefect called bipolaron.

The electronic band structure of a neutral polyacetylene is characterizedby an empty band gap, like in other intrinsic semiconductors. Defect sites(solitons, polarons, bipolarons) can be regarded as electronic states withinthe band gap. The conduction in low-doped polyacetylene is attributedmainly to the transport of solitons within and between chains, as describedby the 'intersoliton-hopping model (IHM)'. Polarons and bipolarons areimportant charge carriers at higher doping levels and with polymers otherthan polyacetylene.

There exist numerous conjugated macromolecules showing high electro-nic conductivities after doping. Materials thus prepared are also termed'synthetic metals' (their significance is documented, for example, by theexistence of a journal of the same name). This section presents only a verybrief summary of this constantly growing field. Further information can befound in numerous reviews and monographs; the most representative onesare listed below.

All important electronically conducting polymers, except perhaps forpolyacetylene, can be prepared electrochemically by anodic oxidation of themonomers. The reaction is initiated by splitting off two hydrogen atomsfrom the monomer molecule (H—M—H), which subsequently polymerizesby interconnecting thus activated sites:

nH—M—H-+ H—(M)*—H + {In - 2)H+ + (2w - 2)e" (5.5.39)

According to this scheme, the consumption of positive charge should beclose to 2F/mol of monomer. Experimental values for various monomersare considerably higher, in the range from 2.1 to 2.7. This is due to the factthat the dehydrogenation of the monomer generally occurs at more positivepotentials than the p-doping of the polymer thus prepared. Therefore, theproduced polymer is oxidized immediately after its formation (p-doped)with the consumption of a non-stoichiometric, additional positive charge:

H—(M)n—H + nyAT-+ H—(M)7+—W-nyAT + nye~ (5.5.40)

where y is the so-called doping level (y —0.1-0.7).Both the mentioned reactions proceed in parallel. The mechanism of

electrochemical polymerization is not completely elucidated, but obviouslythe first step of the polymerization, i.e. the formation of a dimer,H—M—M—H, is rate-determining. As soon as the dimer is formed, itfurther oxidizes more easily to the trimer and higher oligomers, since theoxidation potential of the oligomer gradually decreases with increasingpolymerization degree, n. The polymerization reaction is, therefore, ac-celerated at a constant electrode potential as the polymer chain grows.Several examples of conducting polymers that can be electrochemicallydoped are quoted in Table 5.3.

326

Table 5.3 Examples of electronically conducting polymers, y is thelevel of electrochemical doping and K is the maximum electricalconductivity. Except for polyacetylene and polyparaphenylene, only

p-doping is considered

Polymer Repeat unit y K (S/cm)

Polyacetylene — CH=CH—n-doped 0.09 100p-doped 0.05 1000

Polyparaphenylenen-doped ~ \ 7~ 0.44 100p-doped V = y 0.1 100

Polypyrrole _JI J _ 0.4 500

Polyaniline —M V - N H - 0.2

Polythiophene — ^ JL_ 0.2 190

The monomer and lower oligomers are soluble in the electrolyte, but withincreasing polymerization degree the solubility decreases. After attainingsome critical value, an insoluble film is formed on the anode. Lower(soluble) oligomers can also diffuse from the electrode into the bulk of theelectrolyte, hence the faradaic yield of electrochemical polymerization is, atleast in the primary stages, substantially lower than 100 per cent.

The ideal electropolymerization scheme (Eq. (5.5.39)) is further compli-cated by the fact that lower oligomers can react with nucleophilic substances(impurities, electrolyte anions, and solvent) and are thus deactivated forsubsequent polymerization. The rate of these undesired side reactionsapparently increases with increasing oxidation potential of the monomer,for example, in the series:

aniline < pyrrole < thiophene < furan < benzene

The nucleophilic reaction with the solvent is of crucial importance.Monomers with lower oxidation potentials (aniline and pyrrole) can easilybe polymerized even in aqueous electrolytes. For monomers with higheroxidation potentials, aprotic solvents must be used, such as acetonitrile

327

propylene carbonate, dichloromethane, etc. (Benzene can be polymerizedonly in rather unusual electrolytes.) Side reactions with the solvent as wellas its own oxidation, however, cannot be excluded even in aproticelectrolytes where they impair the electrochemical properties of theproduct. A more elegant way to polythiophene and similar polymers istherefore polymerization starting from oligomers, such as 2,2'-bithiophene(whose oxidation potential is by about 0.4 V lower than that of thiophene).

The best known electrochemically prepared polymer is polypyrrole. Theoxidation potential of pyrrole is about 0.5 V versus SHE; that of polypyrroleis about —0.3 V versus SHE. The oxidative polymerization and p-dopingproceeds, therefore, rather easy. On the other hand, the n-doping ofpolypyrrole has not been so far realized. The electrochemical polymeriza-tion has been performed with a large selection of electrodes (Pt, carbon,metal oxides) and electrolytes ranging from aqueous solutions to moltensalts. Under suitable experimental conditions, a free-standing film ofpolypyrrole can be stripped off from the anode. This forms a basis of anindustrial production of polypyrrole foils, invented by the BASF company.Polypyrrole foils have been studied as charge storing materials for batteries;further prospective applications range from polymeric sensors and electro-chemic devices to antistatic foils for packing and other use.

Pyrrole derivatives substituted in positions 1-, 3-, or 4- have also beenelectrochemically polymerized (positions 2- and 5- must be free forpolymerization). Besides homopolymers, copolymers can also be preparedin this way. Other nitrogen heterocycles that have been polymerized byanodic oxidation include carbazole, pyridazine, indole, and their varioussubstitution derivatives.

In contrast to pyrrole and carbazole, the electrochemical polymerizationof aniline takes place both at the ring and nitrogen positions. Thepolymerization is initiated in acidic medium by the formation of an anilineradical cation, whose charge is localized mainly on the nitrogen atom. Themonomer units are then connected by C—N bonds in para positions(head-to-tail coupling) as depicted in Fig. 5.34A. This form is termedleucoemeraldine (or leucoemeraldine base) and it is essentially non-conducting in both the base and protonized forms (Fig. 5.34B). Strongoxidation of leucoemeraldine base leads to a non-conductive semiquinonicform called pernigraniline base (Fig. 5.34C). Pernigraniline is a ratherunusual form of poly aniline; it was synthesized in a pure form only recentlyby chemical oxidation.

The common form of polyaniline is a 1:1 combination of alternatingreduced (A) and oxidized (C) units; it is termed emeraldine (or emeraldinebase). The emeraldine base is essentially non-conductive, but its conduc-tivity increases by 9-10 orders of magnitude by treating with aqueousprotonic acids. The conductive form of polyaniline can therefore be roughlydepicted as a 1:1 combination of alternating A and D units.

328

-2n(H«e")

Fig. 5.34 Schematic representation of the oxidation/reductionand acidobasic reactions or polyaniline

5.5.6 Non-electrochemical methods

While methods employing radiaoactive tracer techniques have become aclassical tool for the study of adsorption on electrodes, optical methods forthe study of electrodes and processes occurring on them at an atomic ormolecular level have undergone enormously rapid progress, which ischaracteristic for the contemporary development of electrochemistry.

Most suitable are methods studying the interphase between the electrodeand the electrolyte solution under the same conditions as those employed inelectrochemical methods, i.e. in situ. In the first place methods employingelectromagnetic radiation in the microwave, infrared, visible, ultraviolet,X-ray and y-radiation regions belong to this group. The radiation can passmore or less efficiently through the electrolyte, through the material of theelectrochemical cell (window), and even through the electrode without adecrease in its intensity below the detection limit.

The methods of electron and ion optics (Table 5.4) employing a beam ofslow electrons or ions (with energies up to 104eV) for excitation of thesystem have found many applications in the study of the surfaces of solidelectrodes. The very small mean free path of a beam of slow electrons orions, of the order of 1-10 nm in solids and 100 nm in gases at atmosphericpressure, results in the very high surface sensitivity of these methods. On theother hand, however, its application is limited to an ex situ study of theelectrode surface in an ultra high vacuum, i.e. at pressures below 10~6Pa.Various experimental procedures have been proposed for anaerobic transferof the electrode from the electrolyte solution to an ultra high vacuum, ifnecessary with an intact double layer, but without the bulk electrolyte(electrode emersion). In spite of substantial success, e.g. in the study ofthe adsorbed anions on an immersed gold electrode polarized to variouspotentials, it is apparent that transfer of the electrode from the electrolyte

Table 5.4 Basic methods of surface analysis (e = electron, I = ion, hv = photon, other abbreviations are explained in thetext)

Method

XPSAESEMPHIXEPIXESIMSISSRBS

Excitingparticle

hv

e

I

III

Detectedparticle

ee

hv

hv

III

Informationdepth (nm)

1-30.4-3

103

103

0.3-10.3-13-103

Detectionlimit (at.%)

0.1-10.1-1

0.1

io-5-io3

10"5

0.1

Lateralresolution (jum)

10000.1-10

1

3-1000

1001001-3

Range ofelements

>Na

>Na

Bond Destruc-information tibility

V ;;>( + )

+ + +

330

solution to an ultra high vacuum spectrometer is the most sensitiveoperation in the whole procedure of a surface analysis of electrodes.

Methods employing electromagnetic radiation. The interaction of electro-magnetic waves with the electrode, the electrolyte or the interphasebetween them can be connected with a change in the amplitude, phase,frequency and direction of wave propagation. Evaluation of these changesoften yields very detailed information on the studied object. Some methods(e.g. ESR, absorption or luminescence spectroscopy in the ultraviolet andvisible regions) are more suitable studying electrochemically generatedsubstances in solution, while others (e.g. reflection spectroscopy, ellip-sometry, Mossbauer spectroscopy and X-ray diffraction) are applied to thestudy of changes of the electrode surface or in the immediate vicinity of thesurface. The boundary between the two types of methods is not a sharp one,as procedures have been developed that are suitable for obtaining bothtypes of information (e.g. Raman spectroscopy).

A change in the amplitude of the excitation radiation occurs especiallyduring the non-elastic interaction of photons with a substance, connectedwith the transfer of a defined quantum of energy; methods measuring thischange are termed spectroscopic methods. Relatively, the smallest energyquantum corresponds to the energy difference between two states differingin the spin of the unpaired electron, whose degeneracy has been removedby the action of an external magnetic field. These processes are studied bymethods termed electron spin {paramagnetic) resonance, ESR (or EPR). Ofsubstances with an unpaired electron, transition metal compounds and freeradicals can be studied. Electrochemistry is concerned mainly with the studyof organic radicals formed during redox processes at electrodes or insubsequent chemical reactions, often in non-aqueous solvents. With theESR method, radicals in electrolytes at concentrations as low as10~8 mol • dm"3 can be detected in a wide lifetime range. Very unstableradicals can also be studied indirectly, by conversion to a more stableradical by the spin-trapping method. ESR measurements can readily becarried out in situ, and the term SEESR (simultaneous electrochemical ESR)is sometimes used. The shape and hyperfine structure of the ESR lines alsoyields information on the structure of radicals, ion association, solvation,etc. In practice, even the information on the mere fact that the product orintermediate in an electrode process is a radical is valuable and can be ofgreat importance in the discussion of the mechanism of the electrodeprocess. The ESR method has been quite extensively used in electro-chemistry in the study of substances in the electrolyte phase, while lessattention has been paid to the study of paramagnetic sites on the electrodeitself.

Electrochemically generated products can be readily characterized by insitu measurement of their absorption spectra in the ultraviolet and visibleregions. Optically transparent electrodes (OTEs) prepared from thin layers

331

of metals or semiconductors on glass or polymer supports play a key role inthese investigations. Alternatively, very fine, transparent metal mesh can beused as an electrode. Absorption spectra can be advantageously studied bymeans of a working electrode in a thin-layer system (with a thickness ofabout 0.05-0.5 mm), where electrolysis of all the bulk electrolyte lasts onlya few seconds (thin-layer spectroelectrochemistry).

Under certain conditions, the product of the electrode process can bestudied by using electrochemically generated luminescence. This phenom-enon occurs when the reactants generated at the electrode or in a subsequentchemical reaction have such high free energy that they can interact to formproducts in electronically excited states. If these states are capable ofradiative relaxation, then they can be identified according to their lumines-cence spectra. Reactants for luminescence studies can be generated in twoways, either subsequently on a single electrode polarized, e.g. by arectangular or sinusoidal voltage, or simultaneously, e.g. at a rotating ringdisk electrode. Electrochemically generated luminescence finds wide ap-plication in the study of organic substances and radicals and also of somecomplexes of transition metals (for example, Ru(bpy)3

2+, where bpy =bipyridine). This method is especially useful in determining the mechanismsof electrochemical processes.

The most important methods used in in-situ studies of electrode surfacesare various modifications of reflection spectroscopy in the ultraviolet throughinfrared regions. For electrochemical applications, the specular reflection(at smooth electrode surfaces) is much more important than the diffusereflection from matt surfaces. The reflectivity, R, of the electrode/electrolyte interface is defined by:

R=I/I0 (5.5.41)

where Io is the intensiy of incident radiant energy, and / is the intensity ofthat reflected from the interface (some authors distinguish the term 'reflectivity'for specular reflection and 'reflectance' for diffuse reflection.)

Specular reflection of electromagnetic radiation at the (electrochemical)interface is generally described by Fresnel equations. Supposing the mostsimple case that both the electrolyte and electrode are transparent and differonly in their refractive indexes, nx and n2, the reflectivity for normalincidence of the radiation equals:

R = [(n2-n1)/(n2 + nl)]2 (5.5.42)

and R = 1 for grazing incidence. At other angles of incidence 0 < 6 <90°,the reflectivity of radiation polarized normal and parallel to the plane ofreflection is different, the former increases monotonously from value givenby Eq. (5.5.42) to unity, but the latter goes through minimum at angle 6h

(so called Brewster angle):

0b = arctan {n2lnx). (5.5.43)

332

The approximate picture of a transparent electrochemical interface israther far from the actual situation. Nevertheless, the Fresnel equations arevalid also for absorbing media, if one uses the complex refractive indexes,Nk:

Nk = nk-i(aX/4jz), k = l,2 (5.5.44)

(or is absorption coefficient of the medium and A wavelength). In this case,Eq. (5.5.42) is replaced by

R = [(n- I)2 + n2a2X2/Sjt2]/[(n + I)2 + n2a2X2l%n2\ (5.5.45)

where n=njn2 is the relative refractive index. Equation (5.5.45) dem-onstrates the fact that strongly absorbing materials, like metals, are alsostrongly reflecting, with /?—»1 even at the normal incidence.

The most important situation occurs when a film of different opticalproperties is formed at the electrode surface. In this case, theory predictsthat the R value can be changed, even for non-absorbing films, as a result ofexistence of a third phase with different refractive index interspacedbetween the electrode and electrolyte. Therefore, the entire observeddecrease in reflectivity R is not necessarily caused by the absorption ofradiation in the film. This approximation, is, however, reasonably accept-able when the film is supported by a highly reflective phase, such as smoothmetal electrode.

In general, two basic experimental arrangements of reflective spectro-electrochemistry are possible:

(a) The excitation beam passes first through a window in the electrochemi-cal cell, then through the electrolyte and is finally reflected from thesurface of the electrode back into the electrolyte phase (specularexternal reflectance). There is a sufficient number of suitable sub-stances for the preparation of electrodes, electrolytes and windows forworking in the ultraviolet and visible regions. More problems areencountered in the infrared region, as practically all electrolytes absorbinfrared radiation so strongly that external reflection spectra can bestudied only in cells with a very thin (1-10 \xm) layer of electrolytebetween the window (usually Si, Ge, ZnSe, CaF2) and the electrode.

(b) The excitation beam passes through the electrode material and isreflected once or twice from the interface with the electrolyte back intothe electrode material (internal or attenuated total reflection). Thismethod requires the use of an optically transparent electrode. Internalreflectance of infrared radiation can be studied, for example, on agermanium electrode; other materials such as carbon and variousmetals have such high absorption coefficients for infrared radiation thatthey can be used in only very thin layers (5-30 nm) fixed to transparentsupport materials. Vacuum deposited gold layer on a Ge electrode isthe most common arrangement. In practice, a compromise between theelectrical conductivity and infrared absorbance must always be found.

333

The relative changes in reflectance at the electrode are usually very small,typically of the order of 10~4-10~2 per cent, and thus a special techniquemust be used to measure these differences; this is based primarily on signalmodulation and lock-in detection. The simplest modulation method is basedon mechanical interruption of the excitation light beam; this approach has adisadvantage in that it in no way increases the surface sensitivity ofreflectance spectroscopy. Thus it is preferable to employ signal modulationby periodic changes in the electrode potential (electrochemically modulatedreflectance spectroscopy). A further approach to modulation involvesapplication of polarized excitation radiation. This modulation depends onperiodic changes in the polarization plane of the radiation between thedirections perpendicular to and parallel to the plane of reflection of thebeam. This method also increases the surface sensitivity of reflectancespectroscopy as a result of application of specific selection rules for thereflectance of polarized light from the electrode surface. In addition, themeasurement is carried out at constant potential and the information on theelectrode surface is not distorted by changes that could occur duringpotential modulation.

The third approach to reflectance studies at the electrochemical interfacepresents Fourier transformation spectroscopy. This method is used in theinfrared spectral region, where standard commercial instruments (with onlyminor laboratory modifications) are readily applicable. The most commonexperimental technique is known as SNIFTIRS (subtractively normalizedinterfacial Fourier transformation infrared spectroscopy). It is based on themeasurement of external reflectance, /?w at a working potential of theelectrode, £w, and is compared to external reflectance, R, at a potential E,where no significant charge transfer reaction takes place. The plot of(Rv/-R)/R = AR/R versus wavenumber contains therefore only the char-acteristic features introduced by the potential change, whereas thepotential-independent infrared absorptions (electrolyte window, and vibra-tional structures intact by the charge transfer reaction) are numericallycancelled. Negative-going peaks correspond to the increase in concentrationof species generated at the working potential Ew and vice versa.

Reflectance spectroscopy in the infrared and visible ultraviolet regionsprovides information on electronic states in the interphase. The externalreflectance spectroscopy of the pure metal electrode at a variable potential(in the region of the minimal faradaic current) is also termed 'electroreflec-tance'. Its importance at present is decreased by the fact that no satisfactorytheory has so far been developed. The application of reflectance spectro-scopy in the ultraviolet and visible regions is based on a study of theelectronic spectra of adsorbed substances and oxide films on electrodes.

Infrared reflectance spectroscopy provides information on the vibrationalstates in the interphase. It can be interpreted in terms of molecularsymmetry, force constants and chemical bond lengths. The intensity of thespectral peaks of the adsorbed molecules is determined both by standard

334

selection rules and also by specific surface selection rules, differentiatingvarious directions of the polarization of radiation and molecular orientationwith respect to the surface of the electrode. Infrared reflectance spectros-copy has been used, for example, to study the structure of water in theelectrical double layer, adsorption of H2 and CO on a platinum electrodeand the adsorption of intermediates in the oxidation of organic substanceson Pt, Au, Rh and Pd electrodes. The spectra can be interpreted in terms ofthe symmetry changes in the molecules in the electrochemical interface,bonding to the electrode surface, aggregation, etc.

An example of the SNIFTIRS study of a Pt electrode in contact with 0.5 MLiClO4 in propylene carbonate (PC) is demonstrated in Fig. 5.35. Thereference potential, E was 2 V versus Li/Li+, the working potential, £w was3.2 V versus Li/Li+. Anodic oxidation of PC is clearly apparent at theseconditions; it is accompanied by the relative intensity drop of the C=Ostretching vibration at 1785 cm"1 (from the intact PC), and increase of theC=O stretching vibration at 1766 cm"1 (from the oxidized PC). Theoxidation products are further characterized by several new negative goingpeaks in the 1000-1800 cm"1 region, whereas the content of CH3 groupsremains unchanged by oxidation. The most characteristic oxidation productof PC is molecular CO2 identified at 2342 cm"1. The onset of PC oxidation(CO2 formation) was detected by SNIFTIRS already at 2.1 V versus Li/Li+,which indicates that propylene carbonate is much less stable againstoxidative decomposition than it was formerly believed on the basis ofelectrochemical methods (anodic currents at the platinum electrode in thementioned electrolyte are practically negligible in the interval 2-4 V versus

1

0

1

1

- -CH 3_ — - * * • '

C02

' ) C = 0

1

f1

)C=0

PCCH3-CH-CH2 -

: 6wo *C

c

t]\

oxid. PC

5000 2000

Wavenumber/cm-1

1000

Fig. 5.35 SNIFTIRS spectrum from a polished Pt electrodein 0.5 M LiClO4 in propylene carbonate. Reference potential:2.00 V versus Li/Li+ electrode; working potential 3.20 V

versus Li/Li+ electrode. According to P. Novak et al.

335

The absorption of radiation at the electrochemical interface can bestudied by a number of methods—not only optically by the reflection ortransmission technique but also by direct measurement of the energyreleased during radiationless relaxation of excited states. The simplestversion is based on measuring temperature changes in the region around anintermittently irradiated electrode—called photothermal spectroscopy.Alternatively, periodic temperature changes of the electrode can bemonitored in terms of periodic changes in its dimensions; these can bemeasured with a piezoelectric detector. The most commonly employedmethod involves measurement of the intensity of an acoustic signal formedby periodic pressure changes in the region around the intermittentlyirradiated electrode—termed photoacoustic (or optoacoustic) spectroscopyor PAS (OAS). In this method, the excitation radiation is modulated withan acoustic frequency of 101-103Hz, and detection is carried out with amicrophone. The photothermal and photoacoustic spectroscopic methodshave a great advantage over reflectance and transmission techniques in theirminimal requirements on electrode preparation; the electrode can beopaque, can have a rough surface, etc. Photoacoustic spectroscopy andrelated techniques can be used from the ultraviolet to infrared regions; onlythe ultraviolet and visible regions have so far been used in electrochemicalapplications. Interpretation of the spectra is similar to that for absorption orreflectance spectroscopy in the same region. Specific applications include astudy of the efficiency of photoelectrochemical conversion of energy atsemiconductor electrodes.

The reflectance of light waves from the electrode surface involves achange not only in the amplitude but also in the phase of the reflectedradiation. The method termed ellipsometry is based on the fact that, duringreflection from the electrode, linearly polarized light is changed intoelliptically polarized light. The polarized light incident on the electrodesurface can be considered as consisting of two superimposed light waveswith the same frequency and phase but polarized perpendicularly andparallel to the plane of incidence. After reflection from the surface, thesetwo waves are, in general, out of phase and their amplitudes are changedcompared to those of the incident radiation. The interference of these tworeflected waves produces elliptically polarized light or, in special cases(waves with the same amplitude and phase-shifted by JT/2), circularlypolarized light. The phase shift and the ratio of the amplitudes and thus thecorresponding optical parameters of the surface can be found experimen-tally. The capability to determine the thickness of surface oxide films onelectrodes and the kinetics of their growth in situ is most important inelectrochemical applications of this method. Surface films can be studiedwith sufficient sensitivity in a wide range of thicknesses, starting withsubmonolayers. Ellipsometry was also applied to the study of the electricaldouble layer.

Raman spectroscopy is certainly one of the most important opticalmethods employed in electrochemistry. This method is based on an analysis

336

of non-elastically scattered radiation in the ultraviolet and visible regions.Interaction with vibrational states in the scattering medium leads to a slightdecrease (or increase) in the frequency of the scattered radiation comparedto the incident radiation. As a result of specific selection rules, this methodyields structural bonding data complementary to the data obtained by othervibrational spectroscopic methods, especially infrared. The low intensity ofthe scattered light and the low shift of the frequency make it essential thatthe excitation radiation have very precisely defined frequency and highintensity; at present, lasers are exclusively used for this purpose. When thefrequency of the excitation radiation is identical with that of the electronicabsorption band of the studied molecule, a marked increase in the intensityof the Raman effect is observed, exploited in Raman resonance spectro-scopy. Compared to infrared spectroscopy, the use of Raman spectroscopyin electrochemistry is favoured by the high transparence of the electrolyte inthe ultraviolet and visible regions, enabling measurements in situ.Compared to electronic spectra in the ultraviolet and visible regions, Ramanspectra of electrochemically generated substances are characterized by highchemical selectivity and thus far greater capabilities in the study ofmechanistic and kinetic problems. An optical multichannel analyser can beused to study spectra with a high time resolution, i.e. at various stagesduring the change of the electrode potential.

A study of the adsorption of pyridine on a silver electrode firstdemonstrated that, under suitable conditions, Raman spectroscopy can evenbe sufficiently sensitive for the study of substances adsorbed on electrodes,where a modification called surface enhanced Raman spectroscopy (SERS)was used. The adsorption of various molecules on a silver electrode isaccompanied by an enormous increase in the intensity of the Ramanscattered radiation—by up to 106-fold compared to the intensity in thehomogeneous phase. The surface enhancement effect led to great interestduring the 1970s among theoreticians and experimentalists, includingnon-electrochemists. In spite of considerable effort, no satisfactory theoreti-cal basis for this effect has yet been found. The amplification factor dependson the wavelength of the exciting radiation and also very strongly on theelectrode material. In addition to the silver electrode, which is mostly used,copper (an increase of about 105) and gold (about 104) electrodes are alsosuitable for these studies. In addition, some silver alloys or other materialscan also be used, whose surface has been modified by a small amount ofsilver. Increases in the Raman scattering radiation have also been reportedfor Ni, Pd, Cd, Hg, Pt and Al electrodes.

Methods employing X-rays and y-radiation are used less often in electro-chemistry. The possibility of using X-ray diffraction for in situ study of theelectrode surface was first demonstrated in 1980. This technique has long beenused widely as a method for the structural analysis of crystalline sub-stances. Diffraction patterns that are characteristic for the electrochemicalinterface can be obtained by using special electrochemical cells and elec-

337

trodes. The electrode can also be studied in situ by measuring absorption ofX-ray radiation. The extended X-ray absorption fine structure (EXAFS)method is based on the excitation of core-level electrons and, in addition toelemental analysis, permits study of the electronic structure of surfaces. TheEXAFS method requires a quite intense source of X-ray radiation,preferably a synchrotron.

Mossbauer spectroscopy can be used for in situ study of electrodescontaining nuclei capable of resonance absorption of y radiation; forpractical systems, primarily the 57Fe isotope is used (passivation layers oniron electrodes, adsorbed iron complexes, etc.). It yields valuable informa-tion on the electron density on the iron atom, on the composition andsymmetry of the coordination sphere around the iron atom and on itsoxidation state.

Methods employing electron and ion beams. Of the methods employing abeam of electrons or ions for excitation of the sample and/or detection ofthe signal, various surface analysis methods are of greatest importance.Qualitative and semiquantitative elemental analysis of the electrode surfaceis more or less routine for practically all the elements in the periodic system.Nonetheless, the individual methods vary, among other factors, in theirsensitivity for various elements, information depth (i.e. the thickness of thesample layer yielding a detectable signal) and lateral resolution. The latter isdetermined by the surface area of the sample irradiated by the excitationbeam. With a beam of charged species, sufficiently intense radiation can beobtained even for focusing to very small cross-sections, which is theprinciple of microanalysis (microprobes). Scanning of the surface of thesample by the excitation beam yields the distribution of the elements overthe sample surface (scanning microanalysis).

The basic methods of surface analysis are listed in Table 5.4. The X-rayphotoelectron spectroscopic (XPS) method, also termed electron spectro-scopy for chemical analysis (ESCA), one of the most frequently usedmethods, is based on the determination of the kinetic energy of theelectrons emitted from the core levels in a photoelectric process. The energyof the core levels is assessed with the help of the kinetic energy of thephotoelectrons. These energies unambiguously characterize the given atom.The measurement is quite precise (with an error of about 10"1 eV) and thusnot only can elemental analysis be carried out but also fine changes in thecore level energies produced by changes in the electron densities due to theformation of chemical bonds, i.e. chemical shifts, can be studied. Photo-electron spectroscopy can indicate, for example, various oxidation states ofa metal in oxide films on the electrode; it is widely used in the study ofchemically modified electrodes and substances adsorbed on electrodes. TheXPS method yields information on several monolayers of atoms in thesurface. In practice, it is often necessary to analyse deeper layers in the

338

sample; the standard procedure involves a destructive technique of ionsputtering, i.e. depth profiling.

Auger electron spectroscopy (AES) is a further important method ofsurface analysis. It is based on the non-radiative relaxation of vacant corelevels, connected with the emission of electrons (the Auger phenomenon).Measurement of the kinetic energy of emitted Auger electrons permitsdetermination of the core level energies including chemical shifts. Thespectra are more difficult to interpret than XPS spectra, as the electronreorganization occurs among three energy levels. Thus, the energy of theAuger electron (in contrast to the energy of a photoelectron) does notdepend on the excitation energy employed. In the AES method, the corelevel can be excited by any type of radiation with sufficient energy;excitation using electrons is used most often and in general permits muchbetter lateral resolution than excitation using photons. Auger electronspectroscopy is also often combined with depth profiling. It is employed inelectrochemistry for similar purposes as XPS, e.g. to study oxide films,adsorption, corrosion, etc.

A complementary phenomenon to the emission of Auger electrons is theemission of characteristic X-ray radiation (radiative relaxation of the vacantcore level). The probability of emission of X-ray radiation is greater forheavier elements and, on the other hand, the probability of the Augerphenomenon increases for lighter elements. The electron microprobe (EMP)employs excitation by characteristic X-ray radiation using a focused beam ofelectrons. This is not a surface method in the strictest sense, as theinformation depth is much greater than for XPS and AES. A disadvantageof electron excitation of X-ray spectra is the relatively high background ofX-ray bremsstrahlung, decreasing the sensitivity of the analysis. Thebremsstrahlung background can be considerably suppressed by excitation ofX-ray spectra using an ion beam {proton-induced X-ray emission, PIXE, orhelion-induced X-ray emission, HIXE).

The impact of an ion beam on the electrode surface can result in thetransfer of the kinetic energy of the ions to the surface atoms and theirrelease into the vacuum as a wide range of species—atoms, molecules, ions,atomic aggregates (clusters), and molecular fragments. This is the effect ofion sputtering. The SIMS {secondary ion mass spectrometry) method dealswith the mass spectrometry of sputtered ions. The SIMS method has highanalytical sensitivity and, in contrast to other methods of surface analysis,permits a study of isotopes. In materials science, the SIMS method is thethird most often used method of surface analysis (after AES and XPS); ithas so far been used only rarely in electrochemistry.

Methods based on the study of the scattering of an ion beam also belongamong techniques for analysis of surfaces that have been used onlyoccasionally in electrochemistry. The ion-scattering spectroscopic (ISS)method studies the scattering of slow ions (with energy up to 1 keV). The

339

scattering of fast ions (with energy of about 1 MeV) has been studied by theRutherford backscattering spectroscopic (RBS) method. The scattering ofions from a surface can be described over the whole energy range as anelastic binary collision. Determination of the energy and angle of thescattered ions yields, in the first place, information on the mass of thesurface atoms, i.e. elemental analysis. The RBS method is especially usefulfor the detection of heavy atoms at an electrode made of a light element. Itis employed for non-destructive analysis of depth concentration profiles.

In addition to the above methods, yielding primarily semiquantitativeelemental analysis of the surface, further methods of electron and ion opticsare suitable for the study of electrochemical systems. The diffraction of slowelectrons (low energy electron diffraction, LEED) is widely employed forstructural investigation of single-crystal electrodes. In contrast to X-raydiffraction, the LEED method yields information only on the two-dimensional surface structure. The LEED method is used to study cleansurfaces and substances adsorbed on crystalline surfaces. An interestingstructural change ('surface reconstruction') is often observed after contact ofthe electrode with the electrolyte. A disadvantage of the LEED methodcompared to X-ray diffraction is the necessity of working ex situ.

The electrochemistry of modified electrodes occasionally employs a ratherspecial method of electron spectroscopy—inelastic electron tunnelling spec-troscopy (IETS)—based on precise measurement of the electrical conduc-tivity of an electrode covered with an oxide layer and possibly with otheradsorbed substances, in dependence on the voltage. The electrons 'see' thelayer of oxide or adsorbed molecules as a potential barrier that can beovercome by tunnelling. At a definite potential, a characteristic change inthe conductivity is observed, resulting from non-elastic interaction of thetunnelling electrons with the vibrational levels in the studied layer of oxideor adsorbates on the electrode. The IETS method thus yields analogousdata to those obtained from the other vibrational spectroscopic methods.The non-elastic tunnelling phenomenon is not limited by the selection rulesof vibrational spectroscopy and thus bands can be observed that are activein both the Raman and infrared spectra. The IETS method has anadvantage in its ability to study very thin surface layers on the electrodesand, last but not least, in the quite simple instrumentation (e.g. it is notnecessary to work in a vacuum). Measurements can be carried out,however, only ex situ and on a limited number of materials. The aluminiumoxide-covered electrode is most advantageous.

As mentioned above, the methods based on detection of electrons or ionsor probing the electrode surface by these particles are generally hand-icapped by the necessity to move the studied electrode into vacuum, i.e. towork ex-situ. There are, however, two important exceptions to this rule:electrochemical mass spectrometry and electrochemical scanning tunnellingmicroscopy.

340

Electrochemical mass spectrometry was recently introduced for directstudy of the products of electrochemical processes. For this purpose, anelectrochemical cell is connected to the vacuum chamber of a massspectrometer by a porous Teflon membrane. This membrane is at theelectrolyte side covered by a layer of porous platinum serving as a workingelectrode. When the pore size is suitably selected, the solvent molecules donot pass through the hydrophobic Teflon membrane, but some moleculesgenerated at the platinum electrode (e.g. H2, O2, NO, NO2, CO2) canpenetrate through the membrane, and are subsequently detected by themass spectrometer. The analytical response is rather fast, typically < 0.5 s,which allows the measurement of mass-potential curves in parallel withusual current-potential diagrams. According to the used potential regime,we can distinguish the mass spectrometric cyclic voltammetry, MSCV (withtriangular potential changes) or differential electrochemical massspectrometryp, DEMS (with square-wave potential changes). The electroche-mical mass spectrometry seems promising, especially for the study of anodicoxidation of organic substances. For instance, the detection of CO2 duringanodic oxidation of propylene carbonate is a suitable complementarymethod to SNIFTIRS (see the example discussed on page 334).

Electrochemical scanning tunnelling microscopy (ESTM) has recentlybeen recognized as a powerful technique for in-situ characterization of thetopography of electrode surfaces extending to the atomic level. This methodwas originally developed for the study of conducting surfaces in vacuum, butis readily applicable also to surfaces in gaseous or liquid environment. Thelatter application was first demonstrated in 1986 on graphite in aqueoussolutions. The principle of ESTM is very simple: it is based on electrontunnelling between the studied electrode surface and a tip of sharplypointed wire, which is placed in close vicinity above the studied surface. Thetunnelling current decreases exponentially with the distance of the tip fromthe surface (roughly by an order of magnitude for every distance increase by0.1 nm). A significant tunnelling current flows therefore only betweenseveral atoms at the apex of the tip and the nearest atoms in the examinedsurface. The exponential decay of the tunnelling current with distanceenables one to measure the vertical position of the tip with a precision in theorder of 0.01 nm. By sweeping the tip across the studied surface, an imageof its topography with a comparable vertical resolution can be obtained (thelateral resolution is about 0.2 nm).

The experimental set-up usually utilizes a piezoelectric tripod as a supportof the tip (Fig. 5.36). This is movable vertically and laterally over theexamined surface; the vertical distance is fixed by a feedback loop to aconstant tunnelling current at each point of the scan. The contours of thesurface are thus visualized by voltage changes needed to move thepiezoelectric tripod to a desired position.

The in-situ ESTM was utilized for studying adsorbed or underpotential-

341

Fig. 5.36 Scheme of a scanning tunnelling microscope

deposited atoms or molecules on metal and semiconductor electrodes,effects of surface reconstruction and electrochemical roughening, etc. Thelatter was demonstrated, for example after successive oxidation andreduction of the electrode surface in aqueous electrolytes, which is widelyused for 'activation' of Au or Pt electrodes. Fundamental ESTM studies areperformed prevailingly with single-crystal electrodes.

The ESTM experiment provides actually five measurable quantities:tunnelling current, / at the applied voltage, Uy and three dimensions, x, y,z. The standard STM can therefore easily be modified by recording the locall-Uy U-Zy.ox I-z characteristics (z is vertical distance of the tip from theelectrode surface). Plot of dl/dU or d//dz versus x and y brings additionalinformation on the electronic and chemical surface properties (local workfunctions, density-of-states effects, etc.), since these manifest themselvesprimarily as I-U dependences. The mentioned plots are basis of thescanning tunnelling spectroscopy (STS).

Other techniques derived from tunnelling microscopy are based on themeasurement of small repulsive forces (down to 10~9N) acting on adiamond tip attached on the gold foil. These forces originate from repulsionof individual atoms between the tip and the examined surface {atomic forcemicroscope, AFM). Unlike standard tunnelling microscopy, the AFM is notconfined only to electrically conducting samples. Other methods based onthe concept of a force microscopy are laser force microscope (LFN) whichworks with attractive rather than repulsive forces, and magnetic force

342

microscope (MFM). The use of these methods in electrochemistry is so faronly marginal. Electrochemical background has, however, a recentlydeveloped scanning ion conductance microscope (SICM). In this case, thesample is immersed in an electrolyte bath and scanned by a fine electrolyte-filled micropipette while measuring the ion current between two silverelectrodes placed in the bath and inside the micropipette. The obtainedresolution (ca. 10 nm) makes possible to monitor, for example, ion currentsflowing through individual channels in biological membranes.

Radioactive tracer techniques. In electrochemistry, the procedure isessentially the same as in studies of chemical reactions: the electroactivesubstance or medium (solvent, electrolyte) is labelled, the product of theelectrode reaction is isolated and its activity is determined, indicating whichpart of the electroactive substance was incorporated into a given product orwhich other component of the electrolysed system participated in productformation. Measurement of the exchange current at an amalgam electrodeby means of a labelled metal in the amalgam (see page 262) is based on asimilar principle.

These techniques are especially useful for studies of the adsorption ofreactants, intermediates and products of electrode reactions. The simplestcase corresponds to adsorption that is so strong that the electrode can beremoved from the solution, rinsed and its activity measured withoutinterference from desorption. When this procedure is impossible, theactivity of the adsorbate can be measured by the 'electrode loweringmethod'. The radioactive counter is placed under the bottom of the cell,which is made of a plastic foil. The electrode can be located at largedistances from the bottom or can be placed so close to the bottom that onlya thin layer of solution remains beneath it. The radioactivity values at thetwo electrode positions permit determination of the adsorbate activity. Thisprocedure can be repeated many times, thus supplying data on the kineticsof the adsorption process.

References

Adams, R. N., Electrochemistry at Solid Electrodes, M. Dekker, New York, 1969.Albery, W. J., Electrode Kinetics, Oxford University Press, 1975.Albery, W. J., and M. E. Hitchman, Ring Disc Electrodes, Clarendon Press,

Oxford, 1970.Aruchamy, A. (Ed.), Photochemistry and Photovoltaics of Layered Semiconductors,

Kluwer Academic Publishers, Dordrecht, 1992.Augustinski, J., and L. Balsenc, Application of Auger and photoelectron spectros-

copy to electrochemical problems, MAE, 13, 243 (1978).Baker, B. G., Surface analysis by electron spectroscopy, MAE, 10, 93 (1975).Bard, J. A., and L. R. Faulkner, see page 290.Besenhard, J. O., and H. P. Fritz, Elektrochemie schwarzer Kohlenstoffe, Angew.

Chem. 95, 954 (1983).

343

Bond, A. M., Modern Polarographic Methods in Analytical Chemistry, M. Dekker,New York, 1980.

Bond, A. M., K. B. Oldham and C. G. Zoski, Ultramicroelectrode, Anal. Chim.Ada, 216, 177 (1989).

Chien, J. C. W., Poly acetylene: Chemistry, Physics and Materials Science, AcademicPress, New York, 1984.

Clavilier, J., R. Faure, G. Guinet, and R. Durand, Preparation of monocrystallinePt microelectrodes and electrochemical study of the plane surfaces cut in thedirection of (111) and (110) planes, J. Electroanal. Chem., 107, 205 (1980).

Electrodics—Experimental techniques, CTE, 9 (1984).Evans, D. H., Doing chemistry with electrodes, Ace. Chem. Res. 10, 313 (1977).Experimental methods of electrochemistry, CTE, 8 (1984).Faulkner, L. R., Chemical microstructures on electrodes, Chem. Eng. News, 27

February 1984, p. 28.Finklea, H. O. (Ed.), Semiconductor Electrodes, Studies in Physical and Theoretical

Chemistry, 55, Elsevier, Amsterdam, 1988.Fleischmann, M., and I. R. Hill, Raman spectroscopy, CTE, 8, 373 (1984).Fleischmann, M., S. Pons, D. R. Rolinson, and P. P. Schmidt, Ultramicroelectrodes,

Datatech Systems, Marganton, N. C , 1987.Furtak, T. E., K. L. Kliewer, and D. W. Lynch (Ed), Non-traditional approaches to

the study of the solid-electrolyte interface, Surface ScL, 101, 1 (1980).Galus, Z., Fundamentals of Electrochemical Analysis, Ellis Horwood, Chichester,

1976.Greef, R. Ellipsometry, CTE, 8, 339 (1984).Hammond, J. S., and N. Winograd, ESC A and electrode surface chemistry, CTE, 8,

445 (1984).Hansen, W. N., Internal reflection spectroscopy in electrochemistry, AE, 9, 1

(1973).Heineman, W. R., F. M. Hawkridge, and H. N. Blount, Spectroelectrochemistry at

optically transparent electrodes, in Electroanalytical Chemistry Vol. 13 (Ed. A. J.Bard), M. Dekker, New York, 1984.

Heyrovsky, J., and J. Kuta, Principles of Polarography, Academic Press, New York,1966.

Hubbard, A. T., Electrochemistry of well-defined surfaces, Ace. Chem. Res., 13,177 (1980).

Justice, J. B. (Ed.), Voltammetry in Neurosciences, Humana Press, Clifton, 1987.Kastening, B., Electron spin resonance, CTE, 8, 433 (1984).Kazarinov, V. E., and V. N. Andreev, Tracer methods in electrochemical studies,

CTE, 8, 393 (1984).Kavan, L., Electrochemical carbonization of fluoropolymers, in Chemistry and

Physics of Carbon (Ed. P. A. Thrower), Vol. 23, M. Dekker, New York, 1991, p.69.

Kinoshita, K., Carbon, Electrochemical and Physicochemical Properties, John Wiley& Sons, New York, 1988.

Kissinger, P., and W. Heinemann (Eds), Laboratory Techniques in ElectroanalyticalChemistry, M. Dekker, New York, 1984.

Kuwana, T., and N. Winograd, Spectroelectrochemistry at optically transparentelectrodes, in Electroanalytical Chemistry (Ed. A. J. Bard), Vol. 7, p. 1, M.Dekker, New York, 1974.

Kuzmany H., M. Mehring, and S. Roth (Eds), Electronic Properties of Polymers{and Related Compounds), Springer Series of Solid State Sciences, Vols 63, 76, 91,107, Springer-Verlag, Heidelberg, 1985-1992.

Linford, R. G., Electrochemical Science and Technology of Polymers, Vols 1 and 2,Elsevier, London, 1987 and 1990.

344

Macdonald, D. D., Transient Techniques in Electrochemistry, Plenum Press, NewYork, 1977.

Mclntyre, J. D. E., Specular reflection spectroscopy of the electrode/solutioninterface, AE, 9, 61 (1973).

Mclntyre, J. D. E., New spectroscopic methods for electrochemical research, inTrends in Electrochemistry (Eds J. O'M. Bockris, D. A. J. Rand and B. J.Welch), p. 203, Plenum Press, New York, 1977.

McKiney, T. M., Electron spin resonance in electrochemistry, in ElectroanalyticalChemistry (Ed. A. J. Bard), Vol. 10, p. 97, M. Dekker, New York, 1977, p. 97.

Morrison, S. R., see page 244.Muller, R. H., Principles of ellipsometry, AE, 9, 167 (1973).Muller, R. H., Recent advances in some optical experimental methods, Electrochim.

Acta, 22, 951 (1977).Murray, R. W., Chemically modified electrodes, in Electroanalytical Chemistry (Ed.

A. J. Bard), Vol. 13, M. Dekker, New York, 1984.Novak, P., P. A. Christensen, T. Iwashita, and W. Vielstich, Anodic oxidation of

propylene carbonate on platinum, glassy carbon and polypyrrole, /. Electroanal.Chem., 263, 37 (1989).

Osteryoung, J., Developments in electroanalytical instrumentation, Science, 218,261 (1982).

Pleskov, Yu. V., and V. Yu. Filinovskii, The Rotating Disc Electrode, ConsultantsBureau, New York, 1976.

Santhanam, K. S. V., and M. Sharon (Eds), Photochemical Solar Cells, Elsevier,Amsterdam, 1988.

Sarangapani, S., J. R. Akridge and B. Schumm (Eds), The Electrochemistry ofCarbon, The Electrochemical Society, Pennington (1983).

Sawyer, D. T., W. R. Heineman, and J. M. Bube, Chemistry, Experiments forInstrumental Methods, John Wiley & Sons, New York, 1984.

Sawyer, D. T., and J. L. Roberts, Experimental Electrochemistry for Chemists,Wiley-Interscience, New York, 1974.

Skotheim, T. A. (Ed.), Handbook of Conducting Polymers, Vols 1 and 2, M.Dekker, New York, 1986.

Snell, K. D., and A. G. Keenan, Surface modified electrodes, Chem. Soc. Rev., 8,259 (1979).

Southampton Electrochemistry Group, see page 253.Trasatti, S. (Ed.), Electrodes of Conductive Oxides, Elsevier, Amsterdam, 1980.Vincent, C. A., F. Bonino, M. Lazari, and B. Scrosati, Modern Batteries, An

Introduction to Electrochemical Power Sources, E. Arnold, London, 1984.Whittingham M. S., and N. G. Jacobson, Intercalation Chemistry, Academic Press,

Orlando, 1982.Wightman, R. M., and D. O. Wipf, Electroanalytical Chemistry (Ed. A. J. Bard)

Vol. 16, M. Dekker, New York, 1990.Yeager, E., and A. J. Salkind (Eds), Techniques of Electrochemistry, Vols 1-3,

Wiley-Interscience, New York, 1972-1978.Zak, J., and T. Kuwana, Chemically modified electrodes and electrocatalysis, /.

Electroanal. Chem., 150, 645 (1983).

5.6 Chemical Reactions in Electrode Processes

Chemical reactions accompanying the electrode reaction can be classifiedfrom three points of view. They can be sufficiently fast so that they proceed

345

in equilibrium or they are slow enough to constitute a rate-controlling step.Another classification is based on their position with respect to the electrodereaction.

5.6.1 Classification

There are an infinite number of combinations here, only some of whichwill be considered (the individual steps are denoted by symbols as describedon page 252):

/ 1 + / 2 ^ ± R ^Lt p (CE) (a)

C(a) E(a)

R <= ± M ?=± P (EC) (b) (5.6.1)

E(b) C(b)

R ^ P i + P2 (EC) (c)

C(c)

Case (a) If the chemical reaction preceding the electrode reaction, C(a),and the electrode reaction itself, E(a), are sufficiently fast compared tothe transport processes, then both of these reactions can be considered asequilibrium processes and the overall electrode process is reversible (seepage 290). If reaction C(a) is sufficiently fast and E(a) is slow, then C(a)affects the electrode reaction as an equilibrium process. If C(a) is slow,then it becomes the rate-controlling step (a detailed discussion is given inSection 5.6.3).

Case (b) Reaction E(b) is sufficiently fast; then a chemical reactionsubsequent to the electrode reaction, C(b), is either an equilibrium orrate-controlling step. If E(b) is sufficiently slow, then C(b) has no effectswhatever its rate.

Case (c) A chemical reaction C(c) parallel to the electrode reaction has aneffect only when E(c) is sufficiently fast.

Cases (a) and (c) for relatively slow chemical reactions are particularlyinteresting when the electrode reaction is fast and unidirectional, so that theconcentration of substance R at the surface of the electrode approacheszero. Then characteristic limiting kinetic currents are formed on thepolarization curves.

The third approach to classification considers the interaction of substancesundergoing chemical reactions with the electrode. The reactions consideredin this section are either volume reactions or heterogeneous reactions withweak interaction with the electrode surface; such reactions occur often at

346

mercury electrodes. Reactions with strong chemisorption interaction will beconsidered in Section 5.7.

5.6.2 Equilibrium of chemical reactions

A typical example is protonation preceding the electrode reaction

A + H3O+ ±± AH+ + H2O (5.6.2)

characterized by the dissociation constant K':

[A][H3O+]

K ~ [AH+] ( 5 A 3 )

If substance AH+ participates in an irreversible electrode reaction

AH+ + e ^ A H (5.6.4)

the current density is given by the equation

aFE\)

j = -Fkcony exp y -

where c = cA + cAH+- If aH3O+«K' and therefore cAH+«cA, then Eq.

(5.6.5) assumes the form

/ aFE\ cj = -Fkcony exp ( - -^rJcH3o+ — (5.6.6)

with an overall reaction order of 2.The deposition of a metal from a complex also involves equilibria in

solution between the free metal ions, the complexed ions and thecomplexing agent. The formation of the complex MXZ+ from the metal ionMz+ and complexing agent X is characterized by the consecutive stabilityconstants (Section 1.4.3)

so that the solution contains various proportions of the complexes MX2+,MX|+, • • •, MXz

n+ in addition to the free metal ions. Both the free and the

complexed ions can participate in the electrode reaction. Under equilibriumconditions described by Eqs (5.6.7), the following electrode reactions canoccur according to H. Gerischer:

M + X (5.6.8)

347

Each sort of complex including the free ions has a characteristic electrodereaction rate constant (for example, kcMX., / = 0, 1,. . ., n).

Consider the relatively simple case where the chemical reactions offormation and dissociation of the complexes are sufficiently fast so thatequilibrium among the complexes, free ions and complexing agent ismaintained everywhere in the electrolyte during the electrolysis. The rate ofdeposition of the metal is then determined by the electrode reaction of thecomplex MX,-, for which the product kcMX.cMX. is, at the given potential, thelargest for all the complexes.

5.6.3 Chemical volume reactions

According to the theory developed by R. Brdicka and K. Wiesner, achemical reaction prior to the electrode reaction acts as the rate-controllingstep in the electrode process when the electrode reaction disturbs thechemical equilibrium in solution and renewal of this equilibrium is notsufficiently fast. This effect extends to a certain depth in the solution asconcentration changes resulting from the electrode reaction are propagatedby diffusion. According to J. Koutecky and R. Brdicka, the differentialequation describing diffusion to an electrode with a simultaneous chemicalreaction is given by combination of the second Fick law and the equation forthe reaction rate (this idea first appeared in a paper by A. Eucken).Consider a system of two substances participating in a reversible reaction

B <=• A (5.6.9)

with the equilibrium constant K = k2/kl. The substance A is present in amarked excess over substance B (K»1), so that its concentration ispractically not changed by the electrode process. The substance B un-dergoes an irreversible electrode reaction with the rate constant

kc = k^ exp [ - anF(E - E°')/RT]

(see Eq. 5.2.23). The differential equation and boundary and initialconditions for substance B are

dc d^c— = D — f + Jt^CA-ZCce) (5.6.10)

Cfl CfX

t = 0, x > 0: cB = cA/Kt > 0, x = 0: D(dcB/dx) = kccB

After a sufficiently long time interval (t»l/k1K), a steady state isestablished (dcB/dt = 0), giving

D ^ + k1(cA-KcB) = 0 (5.6.11)

348

with the solution

- ' V f ) ] <5"2>Figure 5.37 depicts the stationary distribution of the electroactive

substance (the reaction layer) for &c—»<». The thickness of the reaction layeris defined in an analogous way as the effective diffusion layer thickness (Fig.2.12). It equals the distance \i of the intersection of the tangent drawn to theconcentration curve in the point x = 0 with the line c = cA/K,

(5.6.13)

(5.6.14)

which describes the polarization curve of a kinetic current, controlled by theelectrode reaction and the steady state of the chemical volume reaction anddiffusion. For kc»^JkxKDy the limiting current density is obtained:

The current density is given by the equation

]\ = zFcA (5.6.15)

•g

10

8

6

4

2

/ ~ i " _

1 I i

C AK

f 1

i

-

-

-

-

I1

-5Distance from the electrode, 10 cm

Fig. 5.37 Steady-state concentration distribution (reaction layer) in thecase of a chemical volume reaction preceding an electrode reaction (Eq.( ) ) \ k k 004 1 5 \(5.6.12)); K = 10\

(qx = 0.04 s"1, D = 10~5cm • s \ \i is the

effective reaction layer thickness

349

This relationship forms the basis for the method of determining the rateconstants of fast chemical reactions from the kinetic current.

A similar procedure can be employed for subsequent and parallelchemical reactions and for higher-order reactions.

The reduction of formaldehyde at a mercury electrode is an example of asystem in which a chemical reaction precedes the electrode reaction.Formaldehyde is present in aqueous solution as the hydrated form (asdihydroxymethane), which cannot be reduced at a mercury electrode. Thisform is in equilibrium with the carbonyl form

According to R. Brdicka and K. Vesely the carbonyl form of formal-dehyde is reduced and the limiting kinetic current is given by the rateof the chemical volume reaction of dehydration. An analogous situationoccurs for the equilibria among complexes, metal ions and complexingagents if the rates of complex formation and decomposition are insuffi-cient to preserve the equilibrium. A simple example is the depositionof cadmium at a mercury electrode from its complex with nitrilotriacetic

HN+^COOH = H3X. Here, a single X3~ species is bound in the

COOHcomplex. According to J. Koryta, equilibrium in solution in the pH rangefrom 3 to 5 can be described by the equation

Cd2+ + HX2~ <= ± CdX" + H+ (5.6.17)

The symbols kf and kd are the rate constants for the formation anddissociation of the complex. The equilibrium constant of reaction (5.6.17) isgiven by the relationship

KC6xK3 = (5.6.18)Kd

where KCdX is the stability constant of the complex and K3 is the thirddissociation constant of the acid. If only free metal ions react in the givenpotential range at the electrode and the rate of the electrode reaction of thecomplex is negligible, then the constants in Eq. (5.6.15) for the limitingkinetic current are given as

k\ = /cdcH3o+

(5.6.19)

where cHX2- is the concentration of the complexing agent, present in excess

350

so that its concentration in the electrolyte does not change during theelectrolysis.

A chemical reaction subsequent to a fast (reversible) electrode reaction(Eq. 5.6.1, case b) can consume the product of the electrode reaction,whose concentration in solution thus decreases. This decreases the over-potential of the overall electrode process. This mechanism was proposed byR. Brdicka and D. H. M. Kern for the oxidation of ascorbic acid, convertedby a fast electrode reaction at the mercury electrode to form dehydro-ascorbic acid. An equilibrium described by the Nernst equation is estab-lished at the electrode between the initial substance and this intermediateproduct. Dehydroascorbic acid is then deactivated by a fast chemicalreaction with water to form diketogulonic acid, which is electroinactive.

If a chemical reaction regenerates the initial substance completely orpartially from the products of the electrode reaction, such case is termed achemical reaction parallel to the electrode reaction (see Eq. 5.6.1, case c).An example of this process is the catalytic reduction of hydroxylamine inthe presence of the oxalate complex of TiIV, found by A. Blazek and J.Koryta. At the electrode, the complex of tetravalent titanium is reduced tothe complex of trivalent titanium, which is oxidized by the hydroxylamineduring diffusion from the electrode, regenerating tetravalent titanium,which is again reduced. The electrode process obeys the equations

(5.6.20)Tim + NH2OH solution > TiIV + OH ' + NH2

The NH2 radical rapidly reacts with excess oxalic acid required to form theoxalate complexes of titanium.

Partial regeneration of the product of the electrode reaction by adisproportionation reaction is characteristic for the reduction of the uranylion in acid medium, which, according to D. H. M. Kern and E. F. Orlemanand J. Koutecky and J. Koryta, occurs according to the scheme

(5.6.21)2UO2

+-^UO22+ + UIV (products)

5.6.4 Surface reactions

Heterogeneous chemical reactions in which adsorbed species participateare not 'pure' chemical reactions, as the surface concentrations of thesesubstances depend on the electrode potential (see Section 4.3.3), and thusthe reaction rates are also functions of the potential. Formulation of therelationship between the current density in the stationary state and theconcentrations of the adsorbing species in solution is very simple for a linearadsorption isotherm. Assume that the adsorbed substance B undergoes an

351

irreversible electrode reaction with the rate constant kc and is simul-taneously formed at the electrode by a reversible heterogeneous reactionfrom the electroinactive substance A (see Eq. 5.6.9). The equilibriumconstant for this reaction is K = k2/kx. In contrast to the system consideredabove, the quantities kx and k2 are the rate constants of surface reactions.The substance A is present in solution and is adsorbed on the electrodeaccording to the linear adsorption isotherm

r A = /?cA (5.6.22)

where FA is the surface concentration and cA is the bulk concentration ofsubstance A, and /? is the adsorption coefficient. At steady state,

- A T B ) = 0 (5.6.23)

so that the current density / and limiting current density jx for A:c—» o° followthe equations

nr kc +kxK

jl/nF=k1pcA (5.6.24)

If relationship (5.2.23) is used for kc and relationship (4.3.49) for /3, then a

-0.4 -0.6 -0.8 -1.0 -1.2 -1.4 -1.6 -1.8

Potential vs.SCE.V

Fig. 5.38 Reduction of 10~3M phenylglyoxylic acid at themercury streaming electrode in acetate and phosphatebuffers containing 1 M KNO3: (1) pH 5.02, (2) pH 5.45, (3)pH 5.85, (4) pH 6.25. The curves 2, 3 and 4 are shifted by0.2 V, 0.4 V and 0.6 V with respect to curve 1. The firstwave is controlled by the surface protonation reaction whilethe second is a direct reduction of the acid anion. (Accord-

ing to J. Koryta)

352

j-E curve with a peak is obtained. This phenomenon is typical for a numberof cases where the protonized electroactive form is formed by a surfacereaction with protons (see Fig. 5.38).

References

Bard, A. J., and L. R. Faulkner, see page 290.Brdicka, R., V. Hanus, and J. Koutecky, General theoretical treatment of

polarographic kinetic currents, in Progress in Polarography (Eds P. Zuman and I.M. Kolthoff), Vol. 1, p. 145, Interscience, New York, 1962.

Heyrovsky, J., and Kuta, see page 343.Gerischer, H., and K. J. Vetter, Reaction overpotential (in German), Z. Physik.

Chemie, 197, 92 (1951).Koryta, J., Diffusion and kinetic currents at the streaming mercury electrode, Coll.

Czech. Chem. Comm., 19, 433 (1954).Koryta, J., Electrochemical kinetics of metal complexes, AEy 6, 289 (1967).Koutecky, J., and J. Koryta, The general theory of polarographic kinetic currents,

Electrochim. Acta, 3, 318 (1961).Southampton Electrochemistry Group, see page 253.

5.7 Adsorption and Electrode Processes

As mentioned in Section 5.1, adsorption of components of the electro-lysed solution plays an essential role in electrode processes. Adsorptionof reagents or products or of the intermediates of the electrode reaction orother components of the solution that do not participate directly in theelectrode reaction can sometimes lead to acceleration of the electrodereaction or to a change in its mechanism. This phenomenon is termedelectrocatalysis. It is typical of electrocatalytic electrode reactions that theydepend strongly on the electrode material, on the composition of theelectrode-solution interphase, and, in the case of single-crystal electrodes,on the crystallographic index of the face in contact with the solution.

Adsorption can also have the opposite effect on the rate of the electrodereaction, i.e. it can retard it. This is termed inhibition of the electrodereaction.

5.7.1 Electrocatalysis

A typical adsorption process in electrocatalysis is chemisorption, charac-teristic primarily for solid metal electrodes. The chemisorbed substance isoften chemically modified during the adsorption process. Then either thesubstance itself or some fragment of it is bonded chemically to theelectrode. As electrodes mostly have physically heterogeneous surfaces (seeSections 4.3.3 and 5.5.5), the Temkin adsorption isotherm (Eq. 4.3.46) issuitable for characterizing the adsorption.

The basic characteristics of electrocatalysis will be demonstrated onseveral examples, in the first place on the electrode processes of hydrogen,

353

i.e. reduction of protons present in a proton donor, primarily oxonium ionor water, and oxidation of molecular hydrogen to form a proton thatsubsequently reacts with the solvent or with the lyate ion (in water formingthe oxonium ion or the water molecule).

The electrochemical evolution of hydrogen has long been one of the moststudied electrochemical processes. The mechanism of this process and theoverpotential involved depend on the electrode material; here the over-potential is defined as the difference between the electrode potential atwhich hydrogen is formed at the given current density and the equilibriumpotential of the hydrogen electrode in the given solution. Indeed Eq.(5.2.32) was formulated by J. Tafel as an empirical relationship for theevolution of hydrogen. The quantity a can also be a function of thehydrogen ion activity and of the composition of the solution in general. Theelectrocatalytic character of this process follows from the marked depend-ence of the constants of the Tafel equation for the cathodic process on theelectrode material (Table 5.5). The cathodic process (and in the oppositedirection, the anodic process) has either an EC or an EE mechanism (twosubsequent electrode reactions, one of the original reactant and the other ofthe intermediate). It consists of two steps, in the first of which adsorbedhydrogen is formed from the reactants (H3O+ or H2O):

H3O+ + e*±Hd + H2O( * * }

This process is often called the Volmer reaction (I). In the second step,adsorbed hydrogen is removed from the electrode, either in a chemicalreaction

2H a d s ^H 2 (5.7.2)

(the Tafel reaction (II)) or in an electrode reaction ('electrochemicaldesorption'),

Hads + H3O+ + e«±H2 4- H2O (5.7.3)

(the Heyrovsky reaction (III)).First, we shall discuss reaction (5.7.1), which is more involved than simple

electron transfer. While the frequency of polarization vibration of the mediawhere electron transfer occurs lies in the range 3 x 1010 to 3 x 1011 Hz, thefrequency of the vibrations of proton-containing groups in proton donors(e.g. in the oxonium ion or in the molecules of weak acids) is of the order of3 X 1012 to 3 x 1013 Hz. Then for the transfer proper of the proton from theproton donor to the electrode the classical approximation cannot beemployed without modification. This step has indeed a quantum mechanicalcharacter, but, in simple cases, proton transfer can be described in terms ofconcepts of reorganization of the medium and thus of the exponentialrelationship in Eq. (5.3.14). The quantum character of proton transferoccurring through the tunnel mechanism is expressed in terms of the

354

Table 5.5 Constants a and b of the Tafel equation and the probable mechanism ofthe hydrogen evolution reaction at various electrodes with H3O

+ as electroactivespecies (tfH3o+ ~ !)• (According to L. I. Krishtalik)

Cathode material

PbTlHgCdInSnZnBiGa(l)

(s)AgAuCuFeCoNiPt (anodically activated)

(large /)(poisoned)

Rh (anodically activated)Ir (anodically activated)ReWMoNb (unsaturated with H)

(saturated with H)Ta (unsaturated with H)

(saturated with H)Ti

—a

1.52-1.561.551.4151.40-1.451.33-1.361.251.241.11.050.900.950.65-0.710.77-0.820.66-0.720.670.55-0.720.05-0.100.25-0.350.47-0.720.05-0.100.05-0.100.15-0.210.58-0.700.58-0.680.920.781.21.040.82-1.01

b

0.11-0.120.140.1160.12-0.130.12-0.140.121.120.110.110.100.120.10-0.140.10-0.120.12-0.130.150.10-0.140.030.10-0.140.12-0.130.030.030.03-0.040.10-0.120.100.110.110.190.150.12-0.018

Mechanism

I(s),IIII(s),IIII(s),IIII(s),IIII(s),IIII(s),IIII(s),IIII(s),IIII(s),IIII(s),IIII(s),IIII(s),IIII(s),IIII(s),IIII(s),IIII(s),IIIUI(s)I,III(s)I(s),?IJI(s)UI(s)I,H(s)I,III(s)I,III(s)I,HI(s)IJII(s)I,III(s)I,III(s)I,III(s)

I indicates the Volmer mechanism (Eq. 5.7.1), II the Tafel mechanism (Eq. 5.7.2) and III theHeyrovsky mechanism (Eq. 5.7.3). The slowest step of the overall process is denoted (s).

transmission coefficient,

* « e x p - — —^r2) (5.7.4)

where m is the mass of the proton or its isotope, h is Planck's constant, (ox isthe vibrational frequency of the bond between the proton and the rest of themolecule of the proton donor, cof is the vibrational frequency of the bondbetween the hydrogen atom and the metal and r is the proton tunnellingdistance. When the hydrogen atom is weakly adsorbed, the vibrationalfrequency of the hydrogen-metal bond w{ is small and the proton tunnelling

355

distance r is large and thus K is very small. Consequently, in the evolution ofhydrogen at metals that adsorb hydrogen atoms very weakly, such as Hg,Pb, Tl, Cd, Zn, Ga and Ag, reaction (5.7.1) is the rate-controlling step. Athigh overpotentials, Eq. (5.3.17) is valid for the dependence of the rateconstant on the potential. At lower overpotentials (at very small currentdensities) barrierless charge transfer occurs (see page 274), as indicated inFig. 5.39.

Distinct adsorption of hydrogen can be observed with electrodes with alower hydrogen overpotential, such as the platinum electrode. This pheno-menon can be studied by cyclic voltammetry, as shown in Fig. 5.40 for apoly crystalline electrode. The potential pulse begins at E = 0.0 V, where theelectrode is covered with a layer of adsorbed hydrogen. When the potentialis shifted to a more positive value, the adsorbed hydrogen is oxidized in twoanodic peaks in the potential range from 0.1 to 0.4 V. At even more positivepotentials, no electrode process occurs and only the current for electrodecharging flows through the system. This is especially noticeable at highpolarization rates. The potential range from 0.4 to 0.8 V is termed thedouble-layer region. At potentials of E > 0.8 V, 'adsorbed oxygen' begins toform, i.e. a surface oxide or a layer of adsorbed OH radicals. This process ischaracterized by a drawn-out wave. Evolution of molecular oxygen starts ata potential of 1.8 V. When the direction of polarization is reversed, theoxide layer is first gradually reduced. This process has a certain activationenergy and occurs at more negative potentials than the anodic process. Thereduction of oxonium ions, accompanied by adsorption, occurs at the samepotentials as the opposite anodic process. If this experiment is carried outon the individual crystal faces of a single-crystal electrode (see Fig. 5.41),

Fig. 5.39 Tafel plot of hydro-gen evolution at a mercurycathode in 0.15 M HC1, 3.2 M KIelectrolyte at 25°C. (According

to L. I. Krishtalik)

356

Electrode potential vs.SHE,V

Fig. 5.40 Cyclic voltammogram of a bright platinum electrode in 0.5 M H2SO4.Geometrical area of the electrode 1.25 x 10~3cm2, periodical triangular potentialsweep (dE/dt = 30 V • s"1), temperature 20°C, the solution was bubbled with

argon. (By courtesy of J. Weber)

then the picture changes quite markedly. The shape of the voltammetriccurve is also affected strongly by the procedure of annealing the electrodeprior to the experiment. There are also very marked differences betweenthe first voltammetric curve and the curve obtained after repeated pulsing.All these features are typical for electrocatalytic phenomena.

It was demonstrated by R. Parsons and H. Gerischer that the adsorptionenergy of the hydrogen atom determines not only the rate of the Volmerreaction (5.7.1) but also the relative rates of all three reactions (5.7.1) to(5.7.3). The relative rates of these three reactions decide over themechanism of the overall process of evolution or ionization of hydrogen anddecide between possible rate-determining steps at electrodes from differentmaterials.

The effect of adsorption on the electroreduction of hydrogen ions, i.e. theVolmer reaction, is strongly affected by the potential difference in thediffuse electrical layer (Eq. 5.3.20).

In the presence of iodide ions, the overpotential at a mercury electrodedecreases, although the adsorption of iodide is minimal in the potentialregion corresponding to hydrogen evolution. The adsorption of iodide

357

Fig. 5.41 A single-crystal sphere of platinum (magnification 100x). It is preparedfrom a Pt wire by annealing in an oxygen-hydrogen flame or by electric currentand by subsequent etching. The sphere is then cut parallel to the face with a re-quired Miller index to obtain a single-crystal electrode. (By courtesy of E. Budevski)

retards the electrode process at other electrodes with large hydrogenoverpotentials, such as the lead electrode.

Hydrogen is evolved from water molecules in alkaline media at mercuryand some other electrodes.

As the adsorption energy increases, the rate of the Volmer reaction canincrease until equilibrium is attained and the rate of the process isdetermined by either the Tafel or the Heyrovsky reaction. However, it ismore probable for kinetic reasons that the Tafel reaction will occur atelectrodes that form a moderately strong bond with adsorbed hydrogen(e.g. at platinum electrodes, at least in some cases). Electrodes that adsorbhydrogen strongly such as tungsten electrodes, are in practice completelycovered with adsorbed hydrogen over a wide range of electrode potentials.

358

The Tafel reaction would require breaking the adsorption bonds to twohydrogen atoms strongly bound to the electrode, while the Heyrovskyreaction requires breaking only one such bond; this reaction then deter-mines the rate of the electrode process.

The isotope effect (i.e. the difference in the rates of evolution ofhydrogen from H2O and D2O) on hydrogen evolution is very important fortheoretical and practical reasons. The electrolysis of a mixture of H2O andD2O is characterized, like in other separation methods, by a separationfactor

where cH/cu is the ratio of the atomic concentrations of the two isotopes.The separation factor is a function of the overpotential and of the electrodematerial. The reacting species are H3O+ and H2DO+ in solutions with lowdeuterium concentrations. The S values for mercury electrodes lie between2.5 and 4, for platinum electrodes with low overpotentials between 3 and 4and, at large overpotentials, between 7 and 8.

The overpotential of hydrogen at a mercury electrode decreases sharplyin the presence of readily adsorbed, weak organic bases (especiallynitrogen-containing heterocyclic compounds). A peak appears on thepolarization curves of these catalytic currents. The hydrogen overpotential isdecreased as oxonium ions are replaced in the electrode reaction by theadsorbed cations of these compounds, BHads

+. The product of the reductionis the BHads radical. Recombination of these radicals yields molecularhydrogen and the original base. The evolution of hydrogen through thismechanism occurs more readily than through oxonium ions. The decrease inthe catalytic current at negative potentials is a result of the desorption oforganic compounds from the electrode surface.

The electrode processes of oxygen represent a further important group ofelectrocatalytic processes. The reduction of oxygen to water

O2 + 4H+ + 4e<=±2H2O (5.7.6)

has a standard potential determined by calculation from thermodynamicdata as +1.227 V. The extremely low exchange current density preventsdirect determination of this value. The simultaneous transfer of fourelectrons in reaction (5.7.6) is highly improbable; thus the reaction mustconsist of several partial processes. The non-catalytic electroreduction ofoxygen at a mercury electrode will now be compared to the catalyticreduction at a silver electrode. J. Heyrovsky demonstrated that the stableintermediate in the reduction at mercury is hydrogen peroxide. Figure 5.42depicts the voltammogram (polarographic curve) for the reduction ofoxygen at a dropping mercury electrode. The first wave corresponds to thereduction of oxygen to hydrogen peroxide and the second to the reduction

359

1.01— i i I i i I

0.0 -0.2 -0.4 -0.6 -0.8

Electrode potential vs.SCE.V

-1.0 -1.2

Fig. 5.42 Electrode reactions of oxygen and hydrogen peroxide atmercury (dropping electrode) and silver (rotating disk electrode): (1)reduction of O2 to H2O2 (first wave) and of H2O2 to H2O (second wave) in1 M CH3COOH, 1 M CH3COONa electrolyte at Hg electrode, (2) reductionof O2 in 0.1 M NaOH at Ag electrode, (3) oxidation of H2O2 to O2 andreduction of H2O2 to H2O in 0 . 1 M NaOH at rotating Ag electrode.

(According to M. Bfezina, J. Koryta and M. Musilova)

of hydrogen peroxide to water. The first of these processes

O2 + 2H+ + 2e«±H2O2 (5.7.7)

has a standard electrode potential of 0.68 V and the second reaction

H2O2 4- 2H+ 4- 2e<=*2H2O (5.7.8)

has a potential of 4-1.77 V. Reaction (5.7.7) occurs at a mercury electrodewith an overpotential in acidic and neutral media and reaction (5.7.8) isalways associated with very high overpotentials. A more detailed analysisrevealed that reaction (5.7.7) corresponds to a multistep process, describedby Eqs (5.2.39) to (5.2.48). The rate is determined by the transfer of the

360

first electron, forming the superoxide anion:

O2 + e*±O2- (5.7.9)

The next reactions are connected with the transfer of an additional electronand of hydrogen ions:

O2 +H+«±HO2

(5.7.10)

These reactions proceed very rapidly, so that the overall reaction cor-responds to the transfer of two electrons. As reaction (5.7.9) is very slow inacid and neutral media, the electrode reaction is irreversible and thepolarization curve does not depend on the concentration of hydrogen ions.In weakly alkaline media, reoxidation of HO2~ begins to occur. At pH > 11,the polarization curve at a dropping mercury electrode becomes reversible.In this way, the process proceeds in water and water-like solvents. On theother hand, for example in carbonate melts, the step following after thereaction (5.7.9) is the slow reaction O2~ + e = O2

2~.The reduction of hydrogen peroxide at a mercury electrode is also a

stepwise electrode reaction, where the second step once again is fast, so thatthe observed electrode reaction corresponds to the transfer of two electrons:

H2O2 + e - ^ OH + OH"

The first step in this reaction is connected with breaking of the O—Obond, with a markedly unsymmetrical energy profile (a — 0.2). Otherreactions connected with a similar bond breakage have an analogous course,e.g. the reduction of cystine at a mercury electrode.

Curve 2 in Fig. 5.42 depicts the reduction of oxygen at a silver rotatingdisk electrode. The first wave, corresponding to almost four electrons, canbe attributed to reaction (5.7.6), being a result of the catalytic reduction ofhydrogen peroxide formed as an intermediate in the presence of surfacesilver oxide. This process occurs quite rapidly at potentials more positivethan the reduction potential of oxygen alone. The peak on the voltammetriccurve is formed because of the reduction and thus disappearance of thecatalytically active surface oxide. A further increase in the current cor-responds to the non-catalytic reduction of hydrogen peroxide, which occursat high overpotentials such as that on mercury.

The behaviour of hydrogen peroxide alone (Fig. 5.42, curve 3) is inagreement with this explanation; the catalytic reduction obeys Eq. (5.7.8) atpotentials more positive than the non-catalytic oxidation. The voltammetriccurve obtained is characterized by a continuous transition from the anodicto the cathodic region. The process occurring at negative potentials is then

361

only the reduction of H2O2, proceeding analogously as in the reduction ofoxygen (curve 2).

The anodic evolution of oxygen takes place at platinum and other noblemetal electrodes at high overpotentials. The polarization curve obeys theTafel equation in the potential range from 1.2 to 2.0 V with a b valuebetween 0.10 and 0.13. Under these conditions, the rate-controlling processis probably the oxidation of hydroxide ions or water molecules on thesurface of the electrode covered with surface oxide:

OH(ads) + e (5.7.12)or

H2O(ads)^ OH(ads) + H+ + e (5.7.13)

This is followed by a fast reaction, either

2OH(ads)^ O(ads) + H2O (5.7.14)or

OH(ads)^ O(ads) + H+ + e (5.7.15)

and, finally,

2O(ads)-»O2 (5.7.16)

At more positive potentials, processes occur that depend on the composi-tion of the electrolyte, such as the formation of H2S2O8 and HSO5 insulphuric acid solutions, while the CIO; radical is formed in perchloric acidsolutions, decomposing to form C1O2 and O2. The formation of ozone hasbeen observed at high current densities in solutions of rather concentratedacids.

5.7.2 Inhibition of electrode processes

Electrode processes can be retarded (i.e. their overpotential is increased)by the adsorption of the components of the electrolysed solution, of theproducts of the actual electrode reaction and of other substances formed atthe electrode. Figure 5.43 depicts the effect of the adsorption of methanolon the adsorption of hydrogen at a platinum electrode (see page 353).

The integral of the anodic current over the range of potentials includingboth peaks of the adsorption of hydrogen, as well as the integralcorresponding to the cathodic current, is equal to the charge Qm = —FTm,where Tm is the maximal surface concentration of hydrogen atoms. Whenanother substance is also adsorbed on the electrode (here, methanol), witha surface concentration that does not change during the potential pulse, thissubstance prevents adsorption of hydrogen at sites that it occupies. If thecurve of the hydrogen peaks is integrated under these circumstances, adifferent Q'm value is obtained. If one molecule of adsorbate occupies onesite for hydrogen adsorption, then the relative coverage of the electrode by

362

-0.1 0.1 0.2 0.3

Electrode potential vs.SHE,V

0.4

Fig. 5.43 Influence of methanol adsorption on the hydrogen adsorptionat a bright platinum electrode in 0.5 M H2SO4. This is the same electrodeas in Fig. 5.40 polarized by a single potential sweep (dE/df = 30 V • s 1 )from E = 0.5V to E = — 0.1V. Concentration of methanol (inmol • dm~3 from bottom to top): 0, 10~4, 10~\ 10~2, 10~\ 5. Methanolwas adsorbed at £" = 0.5 V during 2min (the equilibrium coverage was

achieved), t = 20°C. (By courtesy of J. Weber)

the adsorbed substance is

0 =Qm-Q'm (5.7.17)

The dependence of the relative coverage of the platinum electrode withmethanol on its concentration in solution indicates that the adsorption ofmethanol obeys the Temkin isotherm (4.3.46).

The anodic oxidation of molecular hydrogen dissolved in the electrolyteat a Pt electrode consists of several steps: the transport of hydrogen

363

molecules to the electrode, their dissociation with simultaneous adsorptionof atomic hydrogen on the electrode (reversed Tafel reaction (5.7.2)) and itssubsequent oxidation. The shape of the polarization curve depends onwhich of these processes is slowest. The adsorption of hydrogen is affectedby the adsorption of anions, occurring at positive potentials. The adsorbedanions, especially iodide, decrease both the adsorption energy and themaximal amount of hydrogen adsorbed. The effect of iodide is analogous tothat of substances termed cathodic poisons, such as mercury ions, trivalentarsenic or cyanide ions. The retarding effect of adsorbed anions is especiallyapparent for the anodic ionization of hydrogen dissolved in solution at aplatinum electrode. Figure 5.44 shows the voltammograms for this processat a rotating disk electrode. In the vicinity of the equilibrium potential, thisprocess is controlled by convective diffusion (see Eqs 5.4.20 and 5.4.23); atslightly more positive potentials a limiting current appears that is thenindependent of the potential. At much more positive potentials, the surfaceprocess is retarded by the adsorption of sulphate anions and becomes the

I

0.5 1.0Overpotential,V

1.5

Fig. 5.44 The voltammogram of molecular hydrogen at a rotating brightplatinum disk electrode in 0.5 M H2SO4, pHl = 105 Pa, 25°C. The rotation speed^ ( s 1 ) is indicated at each curve. (According to E. A. Aykazyan and A. I.

Fedorova)

364

rate-controlling step in place of diffusion. The higher the rotation rate of theelectrode, the sooner the transition occurs between the transport-controlledprocess and the surface-controlled process. The current increase at 1.6 Vcorresponds to platinum 'surface oxide' formation.

The electrocatalytic oxidation of lower aliphatic compounds such asalcohols, aldehydes, hydrocarbons and formic and oxalic acid at platinum-group electrodes has a number of typical features. The chemisorptionprocess usually involves the dissociation of several hydrogen atoms from theoriginal molecule and a so-far not completely identified intermediate (oftentermed 'reduced carbon dioxide' for the reactions of methanol, formal-dehyde and formic acid) remains adsorbed on the electrode. At electrodepotentials higher than 0.35 V (versus a hydrogen electrode located in thesame solution and saturated with hydrogen at standard pressure) all thehydrogen atoms formed during chemisorption are oxidized anodically. Theadsorbed intermediate can be oxidized at potentials higher than +0.4 V; thelarger the coverage by the adsorbed intermediate, the higher the thresholdpotential of the anodic oxidation. This phenomenon can be explained by the'pair mechanism' of the anodic process, proposed by S. Gilman. Watermolecules in the vicinity of the adsorbed intermediate are highly polarized,facilitating their oxidation. The OH radicals formed then immediatelyattack this intermediate in a surface reaction. At a constant potential, therate of this process decreases because the number of sites available foroxidation of water gradually diminishes. However, if a pulse of increasingpositive potential is applied to the electrode, then the oxidation of waterresulting in OH radical formation is accelerated by increasing the potentialand the rate of the overall electrode process is enhanced, as shown formethanol in Fig. 5.45. Clearly, the rate of the process at low potentialsdecreases with increasing concentration of methanol owing to the retardingeffect of electrode coverage with the intermediate. In contrast, at higherpotentials, the rate of the process increases with increasing concentration(the oxidation of water is so fast that the overall rate increases withincreasing surface coverage).

So far, several examples have been given of the inhibition of electroca-talytic processes. This retardation is a result of occupation of the catalyti-cally more active sites by electroinactive components of the electrolyte,preventing interaction of the electroactive substances with these sites. Theelectrode process can also be inhibited by the formation of oxide layers onthe surface and by the adsorption of less active intermediates and also of theproducts of the electrode process.

The inhibition of electrode processes as a result of the adsorption ofelectroinactive surfactants has been studied in detail at catalytically inactivemercury electrodes. In contrast to solid metal electrodes where knowledgeof the structure of the electrical double layer is small, it is often possible todetermine whether the effect of adsorption on the electrode process atmercury electrodes is solely due to electrostatics (a change in potential 02)

Electrode potential vs.SHE

Fig. 5.45 Single-sweep voltammogram of oxidation of methanol in 0.5 M H2SO4, with the same electrode andthe same conditions of methanol adsorption as in Fig. 5.43. Concentrations of methanol (mol • dm"3): (1) 0, (2)

10"4, (3) 10-*, (4) 10~2, (5) HT1. (By courtesy of J. Weber)

366

or is the result of some other type of inhibition. The steric effect where thesufficiently close approach of the electroactive substance to the electrode isinhibited is typical for large surface-active molecules. The adsorbed layer ofsurfactant replacing the layer of water molecules covering the surface of theelectrode prevents the reacting species and their solvation shells fromforming a configuration suitable for the electrode reaction (see Section 5.3).This leads to a decrease in the value of the rate constant of the electrodereaction. For example, in the case of an irreversible electrode reaction witha rate constant kc and adsorption of an uncharged surfactant, we have

fcc = fc0(l-O) + fc1 (5.7.18)

where k0 and kY are the rate constants for the electrode reaction atunoccupied and occupied sites of the electrode surface, respectively. For theadsorption of surface-active ions, these quantities are also a function of thesurface coverage 0.

It is very simple to determine the value of 0 = F/Fm for a stronglyadsorbed substance in electrolysis with a dropping mercury electrode. If amuch smaller amount of substance is sufficient for complete electrodecoverage than available in the test solution, then the surface concentrationof the surface-active substance F is determined by its diffusion to theelectrode.

The total amount M adsorbed on the electrode in time t (A is the surfaceof the electrode according to Eq. (4.4.4) and F is the surface excess in molesper square centimetre) is given as

M = 0.85Fm2/V/3 = IDJ^A A At (5.7.19)Jo p\dx/x=0

where Dp(dcp/dx)x=0 is the material flux of the surface-active substance tothe electrode, corresponding to the limiting current density and givenaccording to Eq. (2.7.17) by the relationship

where cp is the concentration of the surface-active substance, Dp is itsdiffusion coefficient and c° is its concentration in the bulk of the solution.The surface concentration at time t is given by the equation deduced by J.Koryta:

F = 0.74c£(Dp01/2 (5.7.21)

If complete coverage of the electrode Fm is practically attained at time 0,then

rm \e) (5.7.22)

367

Time,s

Fig. 5.46 The dependence on time of the instantaneouscurrent / at a dropping mercury electrode in a solution of0 . 0 8 M CO(NH 3 ) 6 C1 3 + 0 . 1 M H 2 S O 4 + 0 . 5 M K2SO4 at theelectrode potential where - / « — / d (i.e. the influence ofdiffusion of the electroactive substance is negligible): (1) inthe absence of surfactant; (2) after addition of 0.08%poly vinyl alcohol. The dashed curve has been calculatedaccording to Eq. (5.7.23). (According to J. Kuta and I.

Smoler)

The value of Fm can be calculated from Eq. (5.7.21) where F approaches Fm

at time t = 6. The instantaneous polarographic current /ads controlled by therate of the irreversible electrode reaction of substance A is as follows:

r m1 / 2 i/ads = zFkcAcOx = 0.85zFkom

2/3t2/3\ 1 - - \cOxL \0/ J

(see Eq. 5.7.18), assuming that kx = 0 and that the current is so small thatdiffusion of the electroactive substance can be neglected; / is the current inthe absence of the surface-active substance. Figure 5.46 shows the timedependence of the instantaneous current in the absence and presence of thesurface-active substance.

ReferencesAnson, F. C , Patterns of ionic and molecular adsorption at electrodes, Ace. Chem.

Res., 8,400(1975).Breiter, M. W., Electrochemical Processes in Fuel Cells, Springer-Verlag, New

York, 1969.

368

Kinetics and mechanisms of electrode processes, CTE, 7 (1983)Frumkin, A. N., Hydrogen overvoltage and adsorption phenomena, Part I,

Mercury, AE, 1, 65 (1961); Part II, Solid metals, AE, 3, 267 (1963).Gilman, S., The anodic film on platinum electrode, in Electroanalytical Chemistry

(Ed. A. J. Bard), Vol. 2, p. I l l , M. Dekker, New York, 1967.Heyrovsky, J., and J. Kuta, see page 343.Hoare, J., The Electrochemistry of Oxygen, Interscience, New York, 1968.Krishtalik, L. I., Hydrogen overvoltage and adsorption phenomena, Part III, Effect

of the adsorption energy of hydrogen on overvoltage and the mechanism of thecathodic process, AE, 7, 283 (1970).

Krishtalik, L. I., The mechanism of the elementary act of proton transfer inhomogeneous and electrode reactions, /. Electroanal. Chem.y 100, 547 (1979).

Lamy, C , Electrocatalytic oxidation of organic compounds on noble metals inaqueous solutions, Electrochim. Acta, 29, 1581 (1984).

Marcus, R. A., Similarities and differences between electron and proton transfers atelectrodes and in solution, Theory of hydrogen evolution reaction, Proc.Electrochem. Soc, 80-3, 1 (1979).

Schulze, J. W., and M. A. Habib, Principles of electrocatalysis and inhibition byelectroadsorbates and protective layer, /. Appl. Electrochem., 9, 255 (1979).

Vielstich, W., Fuel Cells, John Wiley & Sons, New York, 1970.

5.8 Deposition and Oxidation of Metals

Sections 5.6.2 and 5.6.3 dealt with the deposition of metals fromcomplexes; these processes follow the simple laws dealt with in Sections 5.2and 5.3, particularly if they take place at mercury electrodes. Thedeposition of metals at solid electrodes (electrocrystallization) and theiroxidation is connected with the kinetics of transformation of the solid phase,which has a specific character. A total of five different cases can bedistinguished in these processes:

1. The deposition occurs at an electrode of a different metal or otherconductive material (e.g. graphite).

2. The deposition occurs at an electrode of the same metal.3. The metal is ionized in the anodic process to produce soluble ions.4. The metal is anodically oxidized to ions that react with the components

of the solution to yield an insoluble compound forming a surface film onthe electrode.

5. A component of the solution participates in the anodic oxidation of themetal, so that the metal is converted directly into a surface film.

These processes either lead to the formation of a new solid phase or theoriginal solid phase grows or disappears. In addition to the electrochemicallaws discussed earlier, this group of phenomena must be explained on thebasis of the theory of new phase formation (crystallization, condensation,etc.).

The basic properties of electrocrystallization can best be illustrated by theexample of the deposition of a metal on an electrode of a different material(case 1).

369

5.8.1 Deposition of a metal on a foreign substrate

The formation of a new phase requires that the system be metastable,such as a supersaturated vapour prior to condensation, a supersaturatedsolution prior to crystallization, etc. The metastable situation of the systemrequired for the formation of a new phase through an electrode reaction isbrought about by the overpotential. At a suitable value of the excess energyin the system nuclei of the new phase are formed. If these nuclei are toosmall, then they become unstable as a result of the relatively high surfaceenergy and decompose spontaneously. If their dimensions exceed a certaincritical value, which is a function of the excess energy of the system, thenthey begin to grow spontaneously. Their formation requires a sufficientlylarge fluctuation of the variables of state of the system or the presence ofsites at the interface at which nuclei can be formed with an overall decreasein the surface energy. The nucleation belongs to those processes where theindividual events (of stochastic nature) occurring on the molecular level canbe observed because of amplifying by subsequent macroscopic processes.

Assume that current is passed either through the total nucleus surfacearea or through part thereof, such as the edge of a two-dimensional nucleusof monoatomic thickness. The transition of the ion Mz+ to the metallic stateobeys the equation for an irreversible electrode reaction, i.e. Eqs (5.2.12),(5.2.23) and (5.2.37). The effect of transport processes is neglected. Thecurrent density at time t thus depends on the number of nuclei and theiractive surface area. If there is a large number of nuclei, then thedependence of their number on time can be considered to be a continuousfunction. For the overall current density at time t we have

-r)vndr (5.8.1)

where A = zFkexp(—azFrj/RT)cMz+ and A is the active surface area of asingle nucleus. The rate of nucleation vn = dNjdt then gives the increase inthe number of nuclei per unit time.

The simplest case occurs when a number of nuclei N° is formed at thebeginning of the process and does not change with time (instantaneousnucleation). This is the case when the electrolysis is carried out by thedouble-pulse potentiostatic method (Fig. 5.18B), where the crystallizationnuclei are formed in the first high, short pulse and the electrode reactionthen occurs only at these nuclei during the second, lower pulse. A secondsituation in which instantaneous nucleation can occur is when the nucleioccupy all the active sites on the electrode at the beginning of theelectrolysis.

For instantaneous nucleation Eq. (5.8.1) assumes the form

(5.8.2)

Hemispherical nuclei will be considered; this case is important for the

370

deposition of liquid metal such as Hg on a platinum or graphite surface. Thesubsequent derivation can be used as an approximation for the general caseof three-dimensional nuclei. The growth of the volume of the nucleus V(t)is described by the equation

(5.8.3)

where Vm is the molar volume of the deposited metal. Simple geometricconsiderations and Eq. (5.8.2) yield

j = -z2fZ V /

This relationship is valid when the growing nuclei do not overlap.Otherwise, the rate of growth of the surface gradually decreases. However,usually the dependence of the current on time in the initial stage is used as adiagnostic criterium for the type of nucleus and nucleation. The simplestcase of progressive nucleation with a constant rate K> Eq. (5.8.1) then gives

. IJTVIKW

The calculation for the important case of two-dimensional nuclei growingonly in the plane of the substrate will be based on the assumption that theseare circular and that the electrode reaction occurs only at their edges, i.e.on the surface, 2nrh9 where r is the nucleus radius and h is its height (i.e.the crystallographic diameter of the metal atom). The same procedure asthat employed for a three-dimensional nucleus yields the following relation-ship for instantaneous nucleation:

J zFT

and for progressive nucleation with constant rate K,

JlhX2Kt2

where Fm is the surface concentration of the metal in the compactmonolayer (mol • cm"2).

In this case the effect of overlapping of the individual nuclei can be simplyexpressed in terms of the statistical Kolmogorov-Avrami theory, where theright-hand side of Eq. (5.8.6) is multiplied by the factor exp(—NlX2t2lz2F2T2

m) and that of Eq. (5.8.7) by the factor (-jTKX2t3/3z2F2T2m).

The properties of the crystal nucleus are a function of its surface energy,its dimensions and the energy of adhesion to the substrate. The equilibriumform of a crystal with the nth plane lying on the substrate is described by

371

the Gibbs-Wulff-Kaischew equation

Yxhl = Y2h2='--={Yn-p)hn (5.8.8)

where y, is the specific surface energy, numerically equal to the surfacetension of the ith crystal plane, hx is the distance of this plane from thecentre of the crystal and j8 is the standard Gibbs energy of adhesion of thenth crystal plane to the substrate. The equilibrium thickness of the crystal(i.e. the distance of the highest point or upper plane of the crystal from thesubstrate) decreases with increasing adhesion energy, until the three-dimensional crystal is finally replaced by a monolayer (i.e. a two-dimensional crystal). If /3>2yn, then the surface layer of adsorbed atoms(Fig. 5.24), termed by M. Volmer ad-atoms ( r < F m ) , is more stable thanthe compact surface of the given metal. Consequently, ad-atoms of themetal can be deposited at a potential that is lower than the electrodepotential of the corresponding metal electrode in the given medium(underpotential deposition). Underpotential-deposited ad-atoms exhibit re-markable electrocatalytic properties in the oxidation of simple organicsubstances.

The nucleation rate constant (Eq. (5.8.5)) K depends on the size of thecritical crystallization nucleus according to the Volmer-Weber equation

(5.8.9)

where the preexponential factor kn is a function of the number of sites onthe substrate where the deposited atom can be accommodated and of thefrequency of collisions of the ion Mz+ with this site. The Gibbs energy forthe formation of a critical spherical nucleus with negligible adhesion energycan be described using the Kelvin equation for the Gibbs energy of a criticalnucleus in the condensation of a supersaturated vapour:

_ 16jtV2my3

G ( 5 8 1 0 )

where y is the surface energy, p is the pressure of the supersaturated vapourand p0 is that of the saturated vapour. The Gibbs energy of supersaturationRT\np/p0 can be replaced by the electrical overpotential energy zFt],yielding

(5.8.11)?>Z F 7]

The rate of three-dimensional nucleation is described in general by theexperimentally verified dependence

—k-ylog K = —f- + k2 (5.8.12)n2

372

while that of two-dimensional nucleation by

log* = l + k2 (5.8.13)

5.8.2 Electrocrystallization on an identical metal substrate

It was mentioned on page 306 (see Fig. 5.24) that, even at roomtemperature, a crystal plane contains steps and kinks (half-crystal posi-tions). Kinks occur quite often—about one in ten atoms on a step is in thehalf-crystal position. Ad-atoms are also present in a certain concentrationon the surface of the crystal; as they are uncharged species, theirequilibrium concentration is independent of the electrode potential. Thehalf-crystal position is of basic importance for the kinetics of metaldeposition on an identical metal substrate. Two mechanisms can be presentin the incorporation of atoms in steps, and thus for step propagation:

(a) The ion M2+ is reduced to an ad-atom that is transported to the step bysurface diffusion and then rapidly incorporated in the half-crystalposition.

(b) The electrode reaction occurs directly between the metal ion and thekink, without intermediate formation of ad-atoms.

In the first case, the rate of deposition depends on the equilibriumconcentration of ad-atoms, on their diffusion coefficient, on the exchangecurrent density and on the overpotential. In the second case, the rate ofdeposition is a function, besides of the geometric factors of the surface,of the exchange current and the overpotential. This mechanism is valid, forexample, in the deposition of silver from a AgNO3 solution.

Formation of subsequent layers can occur either in the way of formationof a single nucleus which then spreads undisturbed over the whole face orby formation of other nuclei before the face is completely covered(multinuclear multilayer deposition). Now let us discuss the first case inmore detail.

The basic condition for experimental study of nucleation on an identicalsurface requires that this surface be a single crystal face without screwdislocations (page 306). Such a surface was obtained by Budevski et al.when silver was deposited in a narrow capillary. During subsequentdeposition of silver layers the screw dislocations 'died out' so that finally asurface of required properties was obtained.

Further deposition of silver on such a surface is connected with a cathodiccurrent randomly oscillating around the mean value

(l)=zFKNmA (5.8.14)

where Nm is the number of atoms per unit surface and A the surface area ofthe electrode. The individual current pulses are shown in Fig. 5.47.

373

PN(r)

Time

B

0 1 2 3 ( 5 6 . 7 N

Fig. 5.47 Deposition of silver on (100) face ofa silver electrode in absence of screw disloca-tions. A—time dependence of current pulses,B—probability distribution of the occurrences

of pulses in the time interval r

Obviously, the nucleation is a randon process which is amplified bysubsequent deposition of many thousands of silver atoms before the surfaceis completely covered (if integrated over the time interval of monolayerformation the current in each pulse corresponds to an identical charge).Such an amplification of random processes is the only way they can beobserved. This situation is quite analogous, for example, to radioactivedecay where a single disintegration is followed, in a Geiger tube, by the flowof millions of electrons.!

In electrochemistry similar phenomena are observed, for example, withthe formation of insoluble films on electrodes or with ion selective channelformation in bilayer lipid membranes or nerve cell membranes (pages 377and 458).

At low overpotentials, the silver electrode prepared according to Budev-ski et al. behaves as an ideal polarized electrode. However, at anoverpotential higher than —6 mV the already mentioned current pulses areobserved (Fig. 5.48A). Their distribution in the time interval r follows thePoisson relation for the probability that N nuclei are formed during the timeinterval r

p(T\ = \(KT)N/N\] exp (—KT) (5.8.15)

The validity of the Poisson distribution for silver nucleation is demonstratedin Fig. 5.48B. The assumption for this kind of treatment is that the nucleusformation is irreversible and that the event is binary consisting of adiscontinuous process (nucleus formation) and a continuous process (flow of

t Because of discontinuous microstructure of all natural systems all processes occurring innature are essentially stochastic (random) and what we observe are only outlines of thephenomena like the average current value in Eq. (5.8.14).

374

Fig. 5.48 A scheme of the spiral growth of a crystal.(According to R. Kaischew, E. Budevski and J.

Malimovski)

time). In the case of silver deposition, the dependence of the nucleation rateconstant K on the overpotential rj follows Eq. (5.8.13).

Another approach to stochastic processes of this kind is the observationof the system in equilibrium (for example, of an electrode in equilibriumwith the potential determining system). The value of the electrode potentialis not completely constant but it shows irregular small deviations from theaverage value E,

E{t) = E+x{t) (5.8.16)

where E is the constant average value and x{i) the fluctuating part, thenoise. For the experimentalist the noise seems to be an unwelcome nuisanceas he is interested in the 'deterministic' average value; however, the'stochastic' data x{t) can surprisingly bring a lot of useful information.

The noise can be described by means of two approaches, namely, in thetime domain and in the frequency domain. For the time domain approachthe characteristic value is the well-known mean square deviation from thevalue over the time T,

= 1/TJ [x(t)-X]2dt (5.8.17)

with Dm = root mean square (rms).The frequency domain description is based on the Fourier integral

transformation of the signal in the time domain into the frequency domain,

Joo= x(t)exp(-ja)t)dt

J(5.8.18)

By this transformation, an important set of values, the power spectrum,

)\2 (5.8.19)

375

is obtained. The analysis of the power spectrum, the aim of which is, forexample, to determine the characteristic constants of the processes fromwhich the noise stems will not be discussed here and the reader isrecommended to inspect the relevant papers listed on page 384.

The electrocrystallization on an identical metal substrate is the slowestprocess of this type. Faster processes which are also much more frequent,are connected with ubiquitous defects in the crystal lattice, in particular withthe screw dislocations (Fig. 5.25). As a result of the helical structure of thedefect, a monoatomic step originates from the point where the newdislocation line intersects the surface of the crystal face. It can be seen inFig. 5.48 that the wedge-shaped step gradually fills up during electro-crystallization; after completion it slowly moves across the crystal face andwinds up into a spiral. The resultant progressive spiral cannot disappearfrom the crystal surface and thus provides a sufficient number of growth

Fig. 5.49 A microphotograph of low pyramid formation during electrocrystalliza-tion of Ag on a single crystal Ag surface. (By courtesy of E. Budevski)

376

sites. Consequently, growth no longer requires nucleation and continues ata low overpotential. Repeated spiral growth forms low pyramids, with asteepness increasing with increasing overpotential. Figure 5.49 depicts aquadratic pyramid on an Ag(100) face. Macroscopic growth on a singlecrystal substrate is called epitaxy.

The kinetics of electrocrystallization conforms to the above descriptiononly under precisely defined conditions. The deposition of metals onpolycrystalline materials again yields products with polycrystalline structure,consisting of crystallites. These are microscopic formations with the struc-ture of a single crystal.

Whiskers are sometimes formed in solutions with high concentrations ofsurface-active substances. These are long single crystals, growing in onlyone direction, while growth in the remaining directions is retarded byadsorption of surface-active substances. Whiskers are characterized by quite

Fig. 5.50 Spiral growth of Cu from a 0.5 M CUSO4 + 0.5 MH2SO4 electrolyte, 25°C, current density 15 mA • cm 2 ,magnification 1250x. (From H. Seiter, H. Fischer, and L.

Albert, Electrochim. Acta, 2, 97, 1960)

377

high strength. From solutions of low concentration metals are oftendeposited as dendrites at high current densities.

Macroscopic growth during electrocrystallization occurs through fastmovements of steps, 10~4-10~5cm high, across the crystal face. Undercertain conditions, spirals also appear, formed of steps with a height of athousand or more atomic layers, so that they can be studied optically (Fig.5.50).

5.8.3 A nodic oxidation of metals

The anodic dissolution of metals on surfaces without defects occurs in thehalf-crystal positions. Similarly to nucleation, the dissolution of metalsinvolves the formation of empty nuclei (atomic vacancies). Screw disloca-tions have the same significance. Dissolution often leads to the formation ofcontinuous crystal faces with lower Miller indices on the metal. Thisprocess, termed facetting, forms the basis of metallographic etching.

Anodic oxidation often involves the formation of films on the surface, i.e.of a solid phase formed of salts or complexes of the metals with solutioncomponents. They often appear in the potential region where the electrode,covered with the oxidation product, can function as an electrode of thesecond kind. Under these conditions the films are thermodynamicallystable. On the other hand, films are sometimes formed which in view oftheir solubility product and the pH of the solution should not be stable.These films are stabilized by their structure or by the influence of surfaceforces at the interface.

Thus films can be divided into two groups according to their morphology.Discontinuous films are porous, have a low resistance and are formed atpotentials close to the equilibrium potential of the corresponding electrodeof the second kind. They often have substantial thickness (up to 1 mm).Films of this kind include halide films on copper, silver, lead and mercury,sulphate films on lead, iron and nickel; oxide films on cadmium, zinc andmagnesium, etc. Because of their low resistance and the reversible electrodereactions of their formation and dissolution, these films are often veryimportant for electrode systems in storage batteries.

Continuous (barrier, passivation) films have a high resistivity (106Q • cmor more), with a maximum thickness of 10~4cm. During their formation,the metal cation does not enter the solution, but rather oxidation occurs atthe metal-film interface. Oxide films at tantalum, zirconium, aluminiumand niobium are examples of these films.

If the electrode is covered with a film, then anodic oxidation of the metaldoes not involve facetting. The surface of the metal either becomes morerough (if the film is discontinuous) or becomes very lustrous (continuousfilms). Films, especially continuous films, retard the electrode reaction ofmetal oxidation. The metal is said to be in its passive state.

Film growth (and its cathodic dissolution) is controlled by similar laws as

378

1 2Time,ms

Fig. 5.51 Successive growth of three calomellayers at an overpotential of 40 mV on a mer-cury electrode. Electrolyte 1 M HC1. (According

to A. Bewick and M. Fleischmann)

metal deposition, i.e. oxidation in half-crystal positions, nucleation andspiral growth, all playing analogous roles. An example is the kinetics offormation of a calomel film on mercury (Fig. 5.51). The time dependence ofthe current obeys approximately the relationship for the progressiveformation of two-dimensional nuclei (see Eq. (5.8.7) and the following texton page 370). However, further nucleation and the formation of a secondand a third layer occur on a partly formed monolayer. Dendrites aresometimes formed during macroscopic growth of oxide layers (Fig. 5.52).This phenomenon is particularly unfavourable in batteries.

Nucleation in continuous films sometimes occurs at less positive potentialsthan those for metal dissolution (Al,Ta) and metal is thus always in apassive state. Again, on other metals (Fe,Ni) nuclei are formed at morepositive potentials than those corresponding to anodic dissolution. Thepotential curve for an iron electrode in 1 M H2SO4 at an increasingly positivepotential has the shape depicted in Fig. 5.53 (the Fe-Fe3+ curve). Theelectrode is first in an active state where dissolution occurs. An oxide filmbegins to form at the potentials of the peak on the polarization curve (calledthe Flade potential, EF) and the electrode becomes passivated; a smallcorrosion current, ;corr, then passes through the electrode. Oxygen is formedat even more positive potentials. The Flade potential depends on the pH ofthe solution and on admixtures in the iron electrode. With increasingchromium content, the Flade potential is shifted to more negative valuesand the current at the peak decreases. This retardation of the oxidation ofthe electrode leads to the anticorrosion properties of alloys of iron andchromium.

The rate-controlling step in the growth of a multimolecular film is thetransport of metal ions from the solution through the film. Diffusion and

379

Fig. 5.52 PbO2 dendrite growth at a lead electrode. Current density0.3 mA • cm"2. (From G. Wranglen, Electrochim. Acta, 2, 130, 1960)

Electrode potential (-E)

oxygenevolution

passivestate

activestate

Fig. 5.53 Anodic processes at an iron electrode. (Accord-ing to K. J. Vetter)

380

migration control this transport when the electric field in the film is not verystrong. If the electrochemical potential of an ion in the film in the steadystate decreases with increasing distance from the surface of the electrode,then the current passing through the film (cf. Eq. 2.5.23) is given as

(5.8.20)

where \ix is the electrochemical potential of the ion at the film-electrolyteinterface, JU2 is the value at the metal-film interface and / is the thickness ofthe film. If the film is insoluble then

j = k2ft (5.8.21)

Solution of the differential equation obtained by combining Eqs (5.8.20)and (5.8.21) yields the Wagner parabolic law of film growth,

I2 = kt (5.8.22)

A strong electric field is formed in very thin films (with a thickness ofabout 10~5cm) during current flow. If the average electrochemical potentialdifference between two neighbouring ions in the lattice is comparable withtheir energy of thermal motion, kT, then Ohm's law is no longer valid forcharge transport in the film. Verwey, Cabrera, and Mott developed a theoryof ion transport for this case.

If the energy of an ion in the lattice is a periodic function of the interionicdistance, then the minima of this function correspond to stable positions inthe lattice and the maxima to the most unstable positions. The maxima aresituated approximately half-way between two neighbouring stable positions.If there is an electric field in the film, A0//, where A0 is the electricpotential difference between the two edges of the film, then a potentialenergy difference of zFaA(p/l is formed between two neighbouring ions. Inthese expressions, z is the charge number and a is the distance between twoneighbouring ions. The energies corresponding to the transfer of an ionfrom two neighbouring rest positions to the energy maximum are theactivation energies for transport of an ion in the direction and against thedirection of the field. Analogous considerations to those employed in thederivation of the basic relationship for electrode kinetics (Section 5.2) yieldthe relationship for the current density, given by the difference between therates of these two processes:

In very strong fields, where the electric energy of the ion is much larger than

381

the energy of thermal motion (zFaA<f>/l »RT), then

( 5 8 2 4 )

which is the exponential law of film growth. On the other hand, ifzFaA<j>/l«RT, then, on expanding the exponentials, we obtain theequation

A l ( 5 8 2 5 )

i.e. Ohm's law. The significance of the constant k follows directly from Eq.(5.8.25):

A?7V(5.8.26)

zFa

where K is the conductivity of the film.

5.8.4 Mixed potentials and corrosion phenomena

As demonstrated in Section 5.2, the electrode potential is determined bythe rates of two opposing electrode reactions. The reactant in one of thesereactions is always identical with the product of the other. However, theelectrode potential can be determined by two electrode reactions that havenothing in common. For example, the dissolution of zinc in a mineral acidinvolves the evolution of hydrogen on the zinc surface with simultaneousionization of zinc, where the divalent zinc ions diffuse away from theelectrode. The sum of the partial currents corresponding to these twoprocesses must equal zero (if the charging current for a change in theelectrode potential is neglected). The potential attained by the metal underthese conditions is termed the mixed potential, Emix. If the polarizationcurves for both processes are known, then conditions can be determinedsuch that the absolute values of the cathodic and anodic currents areidentical (see Fig. 5.54A). The rate of dissolution of zinc is proportional tothe partial anodic current.

In contrast to the equilibrium electrode potential, the mixed potential isgiven by a non-equilibrium state of two different electrode processes and isaccompanied by a spontaneous change in the system. Besides an electrodereaction, the rate-controlling step of one of these processes can be atransport process. For example, in the dissolution of mercury in nitric acid,the cathodic process is the reduction of nitric acid to nitrous acid and theanodic process is the ionization of mercury. The anodic process is controlledby the transport of mercuric ions from the electrode; this process isaccelerated, for example, by stirring (see Fig. 5.54B), resulting in a shift ofthe mixed potential to a more negative value, E'mix.

382

Fig. 5.54 Mixed potential. (A) Zinc dissolution in acid medium. The partialprocesses are indicated at the corresponding voltammograms. (B) Dissolutionof mercury in nitric acid solution. The original dissolution rate characterizedby (1) the corrosion current ja is enhanced by (2) stirring which causes an

383

Processes associated with two opposing electrode processes of a differentnature, where the anodic process is the oxidation of a metal, are termedelectrochemical corrosion processes. In the two above-mentioned cases, thesurface of the metal phase is formed of a single metal, i.e. corrosion occurson a chemically homogeneous surface. The fact that, for example, thesurface of zinc is physically heterogeneous and that dissolution occursaccording to the mechanism described in Section 5.8.3 is of secondaryimportance.

The dissolution of zinc in a mineral acid is much faster when the zinccontains an admixture of copper. This is because the surface of the metalcontains copper crystallites at which hydrogen evolution occurs with a muchlower overpotential than at zinc (see Fig. 5.54C). The mixed potential isshifted to a more positive value, E'm^x, and the corrosion current increases.In this case the cathodic and anodic processes occur on separate surfaces.This phenomenon is termed corrosion of a chemically heterogeneoussurface. In the solution an electric current flows between the cathodic andanodic domains which represent short-circuited electrodes of a galvanic cell.A. de la Rive assumed this to be the only kind of corrosion, calling thesesystems local cells.

An example of the process of a passivating metal is the reaction oftetravalent cerium with iron (see Fig. 5.54D). Iron that has not beenpreviously passivated dissolves in an acid solution containing tetravalentcerium ions, in an active state at a potential of Emix2- After previouspassivation, the rate of corrosion is governed by the corrosion current ja andthe potential assumes a value of Emixl.

For the corrosion phenomena which are of practical interest, the cathodicprocesses of reduction of oxygen and hydrogen ions are of fundamentalimportance, together with the structure of the metallic material, which isoften covered by oxide layers whose composition and thickness depend ontime. The latter factor especially often prevents a quantitative prediction ofthe rate of corrosion of a tested material.

Electrochemical corrosion processes also include a number of processes inorganic chemistry, involving the reduction of various compounds by metalsor metal amalgams. A typical example is the electrochemical carbonizationof fluoropolymers mentioned on p. 316. These processes, that are oftendescribed as 'purely chemical' reductions, can be explained relatively easilyon the basis of diagrams of the anodic and cathodic polarization curves ofthe type shown in Fig. 5.54.

Fig. 5.54 (caption continued)increase of the transport rate of mercuric ions from the metal surface. (C)Effect of copper admixture on zinc dissolution. The presence of copperpatches on zinc surface increases the rate of hydrogen evolution. (D)Oxidation of iron in the solution of Ce4+. On passivated iron the oxidationrate is low (/a) while iron in active state dissolves rapidly (j'a). According to K.

J. Vetter

384

A mixed potential can also be established in processes that do not involvemetal dissolution. For example, the potential of a platinum electrode in asolution of permanganate and manganese(II) ions does not depend on theconcentration of divalent manganese. Anodic oxidation of manganese(II)ions does not occur, but rather anodic evolution of oxygen. The kinetics ofthis process and the reduction of permanganate ions determine the value ofthe resulting mixed potential.

References

Bewick, A., and M. Fleischmann, Formation of surface compounds on electrodes, inTopics in Surface Chemistry (Eds E. Kay and P. S. Bagres), p. 45, Plenum Press,New York, 1978.

Bezegh, A., and J. Janata, Information from noise, Anal. Chem., 59, 494A (1987).Budevski, E., Deposition and dissolution of metals and alloys, Part A, Electrocry-

stallization, CTE, 7, 399 (1983).De Levie, R., Electrochemical observation of single molecular events, AE, 13, 1

(1985).Despic, A. R., Deposition and dissolution of metals and alloys, Part B, Mechanism,

kinetics, texture and morphology, CTE, 7, 451 (1983).Fleischmann, M., M. Labram, C. Gabrielli, and A. Sattar, The measurement and

interpretation of stochastic effects in electrochemistry and bioelectrochemistry,Surface Science, 101, 583 (1980).

Fleischmann, M., and H. R. Thirsk, Electrocrystallization and metal deposition,AE, 3, 123 (1963).

Froment, M. (Ed.), Passivity of Metals and Semiconductors, Elsevier, Amsterdam,1983.

Gilman, S., see page 368.Kokkinidis, G., Underpotential deposition and electrocatalysis, J. Electroanal.

Chem., 201, 217 (1986).McCafferty, E., and R. J. Broid (Eds), Surfaces, Inhibition and Passivation, Ellis

Horwood, Chichester, 1989.Pospisil, L., Random processes in electrochemistry, in J. Koryta et ai, Contem-v poraneous Trends in Electrochemistry (in Czech), Academia, Prague, 1986, p. 14.Stefec, R., Corrosion Data from Polarization Measurements, Ellis Horwood,

Chichester, 1990.Uhlig, H. H., Corrosion and Corrosion Control, John Wiley & Sons, New York,

1973.Vermilyea, D. A., Anodic films, AE, 3, 211 (1963).Vetter, K. J., see page 353.Young, L., Anodic Films, Academic Press, New York, 1961.

5.9 Organic Electrochemistry

As this broad subject has been treated in a great many monographs andsurveys and falls largely in the domain of organic chemistry, the treatmenthere will be confined to several examples.

The tendency of an organic substance to undergo electrochemicalreduction or oxidation depends on the presence of electrochemically active

385

groups in the molecule, on the solvent, on the type of electrolyte andespecially on the nature of the electrode. While oxidation processes at themercury electrode are limited to hydroquinones and related substances,radical intermediates formed by cathodic reduction (e.g. in the potentios-tatic double-pulse method) and some substances forming complexes withmercury, almost all organic substances can be oxidized under suitableconditions at a platinum electrode (e.g. saturated aliphatic hydrocarbons inquite concentrated solutions of sulphuric or phosphoric acid at elevatedtemperatures).

The reduction of organic substances requires the presence of a jr-electronsystem, a suitable substituent (e.g. a halogen atom), an electron gap or anunpaired electron in the molecule.

The electrode reaction of an organic substance that does not occurthrough electrocatalysis begins with the acceptance of a single electron (forreduction) or the loss of an electron (for oxidation). However, thesubstance need not react in the form predominating in solution, but, forexample, in a protonated form. The radical formed can further accept orlose another electron or can react with the solvent, with the base electrolyte(this term is used here rather than the term 'indifferent' electrolyte) or withanother molecule of the electroactive substance or a radical product. Theseprocesses include substitution, addition, elimination, or dimerization reac-tions. In the reactions of the intermediates in an anodic process, thereaction partner is usually nucleophilic in nature, while the intermediate in acathodic process reacts with an electrophilic partner.

According to G. J. Hoijtink, aromatic hydrocarbons are usually reducedin aprotic solvents in two steps:

> 2 -(5.9.1)

The first step is so fast that it can hardly be measured experimentally,while the second step is much slower (probably as a result of the repulsionof negatively charged species, R" and R2", in the negatively charged diffuseelectric layer). The reduction of an isolated benzene ring is very difficult andcan occur only indirectly with solvated electrons formed by emission fromthe electrode into solvents such as some amines (see Section 1.2.3). This is acompletely different mechanism than the usual interaction of electrons fromthe electrode with an electroactive substance.

In the presence of a proton donor, such as water, radical anions accept aproton and the reaction scheme assumes the form

R + H « R H(5.9.2)

RH

386

In the oxidation of aromatic substances at the anode, radical cations ordications are formed as intermediates and subsequently react with thesolvent or with anions of the base electrolyte. For example, depending onthe conditions, 1,4-dimethoxybenzene is cyanized after the substitution ofone methoxy group, methoxylated after addition of two methoxy groups oracetoxylated after substitution of one hydrogen on the aromatic ring, asshown in Fig. 5.55, where the solvent is indicated over the arrow and thebase electrolyte and electrode under the arrow for each reaction; HAcdenotes acetic acid.

J. Heyrovsky and K. Holleck and B. Kastening pointed out that thereduction of aromatic nitrocompounds is characterized by a fast one-electron step, e.g.

—NO2 (5.9.3)

This reduction step can be readily observed at a mercury electrode in anaprotic solvent or even in aqueous medium at an electrode covered with asuitable surfactant. However, in the absence of a surface-active substance,nitrobenzene is reduced in aqueous media in a four-electron wave, as thefirst step (Eq. 5.9.3) is followed by fast electrochemical and chemicalreactions yielding phenylhydroxylamine. At even more negative potentialsphenylhydroxylamine is further reduced to aniline. The same process occursat lead and zinc electrodes, where phenylhydroxylamine can even beoxidized to yield nitrobenzene again. At electrodes such as platinum, nickelor iron, where chemisorption bonds can be formed with the products of the

CH.CN

Et4NCN,Pt

CH30H

KOH,Pt

HAc

KAc,Pt

CN

cyanation

methoxylation

OAcacetoxylation

Fig. 5.55 Different paths of anodic oxidation dependent on electrolyte composi-tion. (According to L. E. Eberson and N. L. Weinberg)

387

reduction of nitrobenzene, azoxybenzene and hydrazobenzene are producedas a result of interactions among the adsorbed intermediates. If azobenzeneis reduced in acid medium to hydrazobenzene, then the product rearrangesin a subsequent chemical reaction in solution to give benzidine.

An example of dimerization of the intermediates of an electrode reactionis provided by the reduction of acrylonitrile in a sufficiently concentratedaqueous solution of tetraethylammonium p -toluene sulphonate at a mercuryor lead electrode. The intermediate in the reaction is probably the dianion

CH2=CHCN + 2 e ^ CH2—CHCN (5.9.4)

which reacts in the vicinity of the electrode with another acrylonitrilemolecule

CH2—CHCN + CH2=CHCN-* NCCHCH2CH2CHCN(5.9.5)

(-) (-)NCCHCH2CH2CHCN + 2H2O-^ NC(CH2)4CN + 2OH~

finally to form adiponitrile (an important material in the synthesis of Nylon606).

Carbonyl compounds are reduced to alcohols, hydrocarbons or pinacols(cf., for example, Eq. 5.1.8), where the result of the electrode processdepends on the electrode potential.

The electrocatalytic oxidation of methanol was discussed on page 364. Theextensively studied oxidation of simple organic substances is markedlydependent on the type of crystal face of the electrode material, as indicatedin Fig. 5.56 for the oxidation of formic acid at a platinum electrode.

M. Faraday was the first to observe an electrocatalytic process, in 1834,when he discovered that a new compound, ethane, is formed in theelectrolysis of alkali metal acetates (this is probably the first example ofelectrochemical synthesis). This process was later named the Kolbe reaction,as Kolbe discovered in 1849 that this is a general phenomenon for fattyacids (except for formic acid) and their salts at higher concentrations. Ifthese electrolytes are electrolysed with a platinum or irridium anode,oxygen evolution ceases in the potential interval between +2.1 and +2.2 Vand a hydrocarbon is formed according to the equation

2RCOO"^ R2 + 2CO2 + 2e (5.9.6)

In addition to hydrocarbons, other products have also been found,especially in the reactions of the higher fatty acids. In steady state, thecurrent density obeys the Tafel equation with a high value of constant6—0.5. At a constant potential the current usually does not depend verymuch on the sort of acid. The fact that the evolution of oxygen ceases in the

388

15-

10-

1 5

o-

1

1

1

Pt(lOO)—

J1

1 1

ftll%

Pt(lOO)

1 1

1

1tPtdio)

rl\Pi(ni)

1

1

1

-0.6 -0 .4 -0 .2 0 0.2

Electrode potential vs.SHE.V0.4

Fig. 5.56 Triangular-pulse voltammograms of oxidationof 0.1 M formic acid in 0.5 M H2SO4 at various faces of asingle-crystal platinum electrode, 22°C, dE/dt =

50 mV • s"1. (According to C. Lamy)

potential range where the Kolbe reaction proceeds can be explained in twoways. The anion of the fatty acid or an oxidation intermediate may blockthe surface of the electrode (cf. page 361), or as a second alternative, theacid may react with an intermediate of anodic oxygen evolution, which thencannot occur. The mechanism of this interesting process has not yet beencompletely elucidated; the two main hypotheses are illustrated in thefollowing schemes:

RCOO--*RCOO a d s -

RCOOa d s"-> RCOOads + e

RCOO a d s->R a d s + CO2 (5-9-7)

Rads"* products, for example R2

389

and

products, for example R2

iRCOCT > R + CO2 + e

(5.9.8)

RCOCT + 2e > R+ + CO2

further productsSimilarly as for other chemical reactions, the activation energy for the

electrode reactions of organic substances depends on the nature andposition of the substituents on the aromatic ring of an organic molecule. Asthe standard potentials are not known for many of these mostly irreversibleprocesses and as they were mainly investigated by polarography, their basicdata is the half-wave potential (Eq. 5.4.27). The difference of the half-wavepotential of the tested substance, i, and of the reference substance (which is,for example, non-substituted) for the same type of electrode reaction (e.g.nitrogroup reduction) is given by the equation

RT k^Ew - £i/2,ref = AE1/2 = AE0' + — - In - ^ (5.9.9)

anF k^fWe assume that neither the preexponential factor of the conditionalelectrode reaction rate constant nor the charge transfer coefficient changesmarkedly in a series of substituted derivatives and that the diffusioncoefficients are approximately equal. In view of (5.2.52) and (5.2.53),

(5.9.10)

where AAHl is the difference between the activation energies of theelectrode reactions of substance i and of the reference substance. R. W.Brockman and D. E. Pearson, S. Koide and R. Motoyama and P. Zumanapplied the Hammett equation, originally deduced for chemical reactions, toelectrode processes. In this concept the activation energy or standard Gibbsenergy of a reaction where members of a homologous series participate isgiven by the relationship

AAHl = 0:0/,(5 9 11)

where py and p) characterize the type of the reaction and ok the substituent.The values of ok which are the same for all reaction types have been listedby L. Hammett. Thus, for the series of half-wave potentials of the sameelectrode reaction type we have

(5.9.12)

390

where p" characterizes the electrode reaction type (reduction of the nitro,aldehydic, chlorine, etc., group) and ok the substituent (mainly in the paraor meta position on the benzene ring).

References

Baizer, M. and H. Lund (Eds), Organic Electrochemistry, M. Dekker, New York,1983.

Fleischmann, M., and D. Pletcher, Organic electrosynthesis, Roy. Inst. Chem. Rev.,2, 87 (1969).

Fry, A. J., Synthetic Organic Electrochemistry, John Wiley & Sons, Chichester,1989.

Fry, A. J., and W. E. Britton (Eds), Topics in Organic Electrochemistry, PlenumPress, New York, 1986.

Kyriacou, D. K., and D. A. Jannakoudakis, Electrocatalysis for Organic Synthesis,Wiley-Interscience, New York, 1986.

Lamy, C , see page 368.Peover, M. E., Electrochemistry of aromatic hydrocarbons and related substances,

in Electroanalytical Chemistry (Ed. A. J. Bard), Vol. 2, p. 1, M. Dekker, NewYork, 1967.

Shono, T., Electroorganic Chemistry as a New Tool in Organic Synthesis, Springer-Verlag, Berlin, 1984.

Torii, S., Electroorganic Syntheses, Methods and Applications, Kodansha, Tokyo,1985.

Weinberg, N. L. (Ed.), Techniques of Electroorganic Synthesis, Wiley-Interscience,New York, Part 1, 1974, Part 2, 1975.

Weinberg, N. L., and H. R. Weinberg, Oxidation of organic substances, Chem.Rev., 68,449(1968).

Yoshida, K., Electrooxidation in Organic Chemistry, The Role of Cation Radicals asSynthetic Intermediates, John Wiley & Sons, New York, 1984.

Zuman, P., Substitution Effects in Organic Polarography, Plenum Press, New York,1967.

5.10 Photoelectrochemistry

5.10.1 Classification of photoelectrochemical phenomena

Photoelectrochemistry studies the effects occurring in electrochemicalsystems under the influence of light in the visible through ultraviolet region.Light quanta supply an extra energy to the system, hence the electrochemi-cal reactions, which are thermodynamically or kinetically suppressed in thedark, may proceed at a high rate under illumination. (There also exists anopposite effect, where the (dark) electrochemical reactions lead to highlyenergetic products which are able to emit electromagnetic radiation. This isthe principle of 'electrochemically generated luminescence', mentioned inSection 5.5.6.) Two groups of photoelectrochemical effects are traditionallydistinguished: photogalvanic and photovoltaic.

The photogalvanic effect is based on light absorption by a suitablephotoactive redox species (dye) in the electrolyte solution. The photo-excited dye subsequently reacts with an electron donor or acceptor process,taking place in the vicinity of an electrode, is linked to the electrode

391

reaction which restores the original form of the dye. The cycle is terminatedby a counterelectrode reaction in the non-illuminated compartment of thecell. Both electrode reactions may (but need not) regenerate the reactantsconsumed at the opposite electrodes. The photopotential and photocurrentappear essentially as a result of concentration gradients introduced by anasymmetric illumination of the photoactive electrolyte solution between twoinert electrodes. Therefore, any photogalvanic cell can, in principle, beconsidered as a concentration cell.

The photovoltaic effect is initiated by light absorption in the electrodematerial. This is practically important only with semiconductor electrodes,where the photogenerated, excited electrons or holes may, under certainconditions, react with electrolyte redox systems. The photoredox reaction atthe illuminated semiconductor thus drives the complementary (dark)reaction at the counterelectrode, which again may (but need not) regeneratethe reactant consumed at the photoelectrode. The regenerative mode ofoperation is, according to the IUPAC recommendation, denoted as'photovoltaic cell' and the second one as 'photoelectrolytic cell'. Alternativeclassification and terms will be discussed below.

The term 'photovoltaic effect' is further used to denote non-electrochemical photoprocesses in solid-state metal/semiconductor inter-faces (Schottky barrier contacts) and semiconductor/semiconductor {pin)junctions. Analogously, the term 'photogalvanic effect' is used moregenerally to denote any photoexcitation of the d.c. current in a material(e.g. in solid ferroelectrics). Although confusion is not usual, electrochemi-cal reactions initiated by light absorption in electrolyte solutions should betermed 'electrochemical photogalvanic effect', and reactions at photoexcitedsemiconductor electodes 'electrochemical photovoltaic effect'.

The boundary between effects thus defined is, however, not sharp. If, forinstance, light is absorbed by a layer of a photoactive adsorbate attached tothe semiconductor electrode, it is apparently difficult to identify thelight-absorbing medium as a 'solution' or 'electrode material'. Photoexcitedsolution molecules may sometimes also react at the photoexcited semicon-ductor electrode; this process is labelled photogalvanovoltaic effect.

The electrochemical photovoltaic effect was discovered in 1839 by A. E.Becquerelt when a silver/silver halide electrode was irradiated in a solutionof diluted HNO3. Becquerel also first described the photogalvanic effect in acell consisting of two Pt electrodes, one immersed in aqueous and the otherin ethanolic solution of Fe(ClO4)3. This discovery was made about the sametime as the observation of the photovoltaic effect at the Ag/AgX electrodes.The term 'Becquerel effect' often appears in the old literature, even fordenoting the vacuum photoelectric effect which was discovered almost 50years later. The electrochemical photovoltaic effect was elucidated in 1955by W. H. Brattain and G. G. B. Garrett; the theory was further developed

t Father of Antoine-Henri Becquerel, the famous French physicist and discoverer ofradioactivity.

392

in detail by H. Gerischer. The photogalvanic effect was first interpreted in1940 by E. Rabinowitch, and more recently studied by W. J. Albery, M. D.Archer and others.

Besides the classification of photoelectrochemical effects (cells) accordingto the nature of the light-absorbing system, we can also distinguish themaccording to the effective Gibbs free energy change in the electrolyte, AG:

1. AG = 0: Regenerative photoelectrochemical cell. The reactants consumedduring the photoredox process (either in homogeneous phase or at thesemiconductor photoelectrode) are immediately regenerated by sub-sequent electrode reactions. The net output of the cell is an electricalcurrent flowing through the cell and an external circuit, but nopermanent chemical change in the electrolyte takes place.

2. AG =£ 0: Photoelectrosynthetic cell. The products of the homogeneous orheterogeneous photoredox process and the corresponding counter-electrode reactions are sufficiently stable to be collected for later use.Practically interesting is the endoergic photoelectrosynthesis (AG >0),producing species with stored chemical energy (fuel). The most impor-tant example is the splitting of water to oxygen and hydrogen. Theexoergic process (AG<0) is also called photocatalytic, since the lightprovides only the activation energy for the reaction, which is otherwisethermodynamically possible even in the dark. Although no chemicalenergy is stored, useful products can sometimes result; an example is thesynthesis of NH3 from N2 and H2.

5.10.2 Electrochemical photoemission

In contrast to semiconductors, metals are usually not active as photo-electrodes owing to the fast thermal recombination of photogeneratedcharge carriers in the metal. The only light-stimulated charge transferprocess at the electrolyte/metal electrode interface is photoemission ofelectrons. It is, in principle, similar to the conventional photoelectric effectin vacuum, i.e. it takes place under illumination with photons of sufficientlyshort wavelengths to overcome the binding energy of electron in the metal.Photoelectrons, emitted from the metal electrode into the electrolytesolution, are rapidly solvated and subsequently captured by a reduciblesolution species. The electrochemical photoemission was studied mostly atmercury electrode, first by M. Heyrovsky. The dependence of the photo-emission current on the radiation frequency differs from that of a conven-tional vacuum photoemission. Gurevich et al. deduced the relationship:

Ic = A(hv-hv0-eE)5/2 (5.10.1)

where /c is the photocurrent, A is a constant, hv is the energy of the lightquantum, hv0 is the threshold energy for photoemission at zero chargepotential, e is the electron charge, and E is the electrode potential.

393

5.10.3 Homogeneous photoredox reactions and photogalvanic effects

The photogalvanic effects are initiated by a homogeneous photoredoxreaction of an electrolyte redox system with a suitable photoexcited organicor organometallic substance (dye), S. The photon absorption produces ashort-lived, electronically excited dye molecule, S*:

whose structure and physicochemical properties are different. The conversionof 5* back to 5 proceeds by various radiative and non-radiative relaxationprocesses, which are characterized by a pseudo-first-order rate constant, kb.Its reciprocal value defines the lifetime of the excited state, r0:

ro=l/kb (5.10.2)

(the fraction of the S* molecules, deactivated during the time r0, equals1 — e~l — 63%). Only excitations of electrons from the highest occupiedmolecular orbital (HOMO) to the lowest unoccupied molecular orbital(LUMO), which produce reasonably long-lived S* molecules are practicallyimportant. This is schematically depicted in Fig. 5.57.

Most of the ground-state molecules contain all the electrons paired. (The0 2 molecule is an important exception to this rule.) The excited moleculemay occur in two states, either with none or two unpaired electrons; thesestates are called excited singlet, 5X, or excited triplet, Tx. The latter state is,according to the Pauli rule, energetically favoured and its lifetime is higherby orders of magnitude, since the reactions with singlet molecules as well asradiative relaxation (phosphorescence) are spin-forbidden.

The photoexcited S*{TX) molecule may have a sufficiently long lifetime todiffuse some distance in the electrolyte solution and to take part in a

eo(vacuum) - -

LUMO

HOMO

S(S0) S*(S1)

Fig. 5.57 Electronic excitation of a molecule

394

bimolecular homogeneous reaction with a molecule acting as electron donor(D) or acceptor (A). In the first case, we speak about reductive quenching:

S* + D ^ S ~ + D + (5.10.3)

and in the second case, about oxidative quenching:

S*+A^S++A~ (5.10.4)

Photoredox reactions involving excited singlet molecules, 5*(5X) are rela-tively rare owing to short lifetimes r0; one exception is the molecule ofoxonine (oxygen analogue of thionine, see below) possessing relatively longsinglet lifetime (2.4 ns) and also small quantum yield of a non-radiativeSl-^ 7i conversion (intersystem crossing), about 3 • 10~3. Oxonin, therefore,reacts photochemically, e.g. with Fe2+ via the excited singlet state.

Figure 5.57 demonstrates that the photoexcited molecule S* exhibits, inboth spin states, a lower ionization potential (IP*) and a higher electronaffinity (EA*) in comparison with that of the ground state 5 molecule (IP,EA). Consequently, the photoexcited molecule 5* is a better oxidizing aswell as reducing agent than the ground state molecule, S. Neglecting theentropy and pressure changes, we can express the relative change of thestandard redox potential of the S/S~ couple as follows:

E0(S/S~) - E0(S*/S~) = -(AG - ^G*)|F - -ejF (5.10.5)

and analogously:

E°(S+/S) - E°(S+/S*) - ejF (5.10.6)

£ex is the excitation energy of 5 (mostly of the first triplet state, S*{TX)). Theactual changes of the redox potentials introduced by photoexcitation aredramatic (cf. Table 5.6). Thus, the photoexcited molecule S* is a powerfulredox reactant to drive various unusual redox reactions in the electrolyte.

Table 5.6 Properties of three typical photoredox-active mole-cules, bpy denotes 2,2' bipyridine, TMPP is tetra N-methylpyridine porphyrin; Amax is the wavelength of the absorp-tion maximum, e is the extinction coefficient at Amax, cpT is thequantum yield of the formation of the excited triplet state, r0 is

its lifetime, and EQ are standard redox potentials

^max

e [cm2iumor1]

<PTr0 [jus]E0(S/S')[V]E0(S+/S)[V]E0(S*/S-)[V]E0(S

+/S*)[V]

Thionine*

599560.557.70.192—1.4—

Ru(bPy)r

452141.00.6

-1.281.260.84

-0.86

ZnTMPP+

560160.90

655—1.2—

-0.6

1 pH = 2 (redox potentials of the system thionine/semithionine).

395

The participation of 5* molecules in homogeneous redox processes ismainly controlled by their short lifetimes. The lifetime of 5* is shortened bya bimolecular redox reaction (5.10.3) or (5.10.4) as follows:

(kT is a rate constant of the quenching reaction and c is a concentration ofthe reactant D or A). Equation (5.10.7) can be rearranged to a morepractical form:

TQ/ T = 1 /v.sv^ v^• J-0. o)

where Ksv = rokr is the so-called Stern-Volmer constant.Figure 5.58 presents a general scheme of photogalvanic cells based on

reductive (a) or oxidative (b) quenching of 5*. These cells are regenerativeif D and A are two forms of the same redox couple (D+ = A and D =A~),otherwise the products D + and A~ accumulate in the electrolyte and the cellis photoelectrosynthetic.

Some properties of three typical photoredox-active dyes are summarizedin Table 5.6. The most important photogalvanic system is based on thephotoredox reaction of thionine with Fe2+. It was first studied by E.Rabinowitch in 1940 and recently optimized by J. W. Albery et al. Thestructural formula of thionine, Thi+ and its reduction products (Sem+,Leu2+) are as follows:

N2N

H,N+ ' NH3+

The photoexcited thionone, *Thi+ is reduced by Fe2 + to the unstablesemithionine (Sem+) which further disproportionates to Thi+ and thedoubly reduced form of thionine, leucothionine (Leu2 +):

Fe2 + + H + ^> Sem+ + Fe3 +*Thi

2Sem+ Leu2+(5.10.9)

(5.10.10)

The anodic reaction (taking place in the illuminated cell compartment) is atwo-electron process regenerating the initial form of Thi+:

Leu2+-* Thi+ + 2e" + 3H+ (5.10.11)

396

transparent electrode

photogalvanic electroly

dark electrode

i

I(a) (b)

Fig. 5.58 Scheme of a photogalvanic cell. The homogeneous photoredox processtakes place in the vicinity of the optically transparent anode (a) or cathode (b)

The two-electron oxidation of the dye is not very common; other dyesusually undergo one-electron redox reactions. The cathodic reaction (takingplace in the non-illuminated cell compartment) regenerates the complemen-tary redox system:

Fe3+ + e"-»Fe2+ (5.10.12)

According to the reactions (5.10.9)-(5.10.12), we can classify this cell as aregenerative (AG = 0) photogalvanic cell based on reductive quenching(Fig. 5.58a). Good performance of the iron-thionine cell is conditioned bythe generation of *Thi+ (and Leu2+) in close vicinity (below ca. 1 jum) ofthe anode; this can be achieved by using an optically transparent (e.g.SnO2) anode as a window (Fig. 5.58). The selectivity of the anode andcathode for the desired reactions (5.9.24) and (5.9.25) may be furtherincreased by selecting the proper electrode materials or by chemicalmodification of the anode.

A regenerative photogalvanic cell with oxidative quenching (Fig. 5.58b) isbased, for example, on the Fe3+-Ru(bpy)|+ system. In contrast to theiron-thionine cell, the homogeneous photoredox process takes place nearthe (optically transparent) cathode. The photoexcited *Ru(bpy)l+ ionreduces Fe3+ and the formed Ru(bpy)3+ and Fe2+ are converted at theopposite electrodes to the initial state.

More complicated photogalvanic cells may employ two light absorbingsystems: one in the cathodic process and the other in the anodic process. An

397

interesting example is a cell with two different chlorophyll-based absorbers.The cathodic process starts by photoexcitation of a modified naturalchloroplast with deactivated photosystem II (PS II—for terminology ofphotosynthetic processes see page 469). The remaining PS I reduces, forexample, anthraquinone-2-sulphonate. The regenerative cycle of PS I islinked to tetramethyl-p-phenylenediamine which is cycled at the cathode.The complementary anodic process is O2 evolution by the action of a secondchloroplast system with active PS II; electrons from photoexcited PS II aretransferred to the tetramethyl-p-phenylenediamine and the cycle is termin-ated by its oxidation at the anode. The whole cell resembles photosynthesis,the net output being O2 and reduced anthraquinone-2-sulphonate (cf.Section 6.5.2).

The efficiency of the photocurrent generation in practical photogalvaniccells is generally low, since it is limited by short lifetimes of the exciteddyes, parasitic electron transfer reactions, etc.

The importance of the photogalvanic effect rests in basic electrochemicalresearch of homogeneous photoredox reactions. To this purpose, originalexperimental techniques have been developed, e.g. voltammetry with anoptically transparent rotating disk electrode. The photoexcitation of reac-tants makes the study of redox reactions possible in a considerably broaderregion of the Gibbs free energy changes than with the ground-statereactants. This brings interesting conclusions, for example, for the formula-tion of a more general theory of electron transfer, preparation of com-pounds in less-common oxidation states, etc.

5.10.4 Semiconductor photoelectrochemistry and photovoltaic effects

Basic properties of semiconductors and phenomena occurring at thesemiconductor/electrolyte interface in the dark have already been discussedin Sections 2.4.1 and 4.5.1. The crucial effect after immersing thesemiconductor electrode into an electrolyte solution is the equilibration ofelectrochemical potentials of electrons in both phases. In order to quantifythe dark- and photoeffects at the semiconductor/electrolyte interface, acommon reference level of electron energies in both phases has to bedefined.

Let us choose, as an arbitrary reference level, the energy of an electron atrest in vacuum, e0 (cf. Section 3.1.2). This reference energy is obvious instudies of the solid phase, but for the liquid phase, the Trasatti's conceptionof 'absolute electrode potentials' (Section 3.1.5) has to be adopted. Theformal energy levels of the electrolyte redox systems, £REDOX, referred to£(), are given by the relationship:

£REDOX = " ^ R E D O X - kT In (cox/cred) + £SHE (5.10.13)

where £REDOX is the standard redox potential of the particular redox systemand cox, cred the concentrations of the oxidized and reduced forms,

398

respectively. £SHE is the energy level of the standard hydrogen electrodereferred to e0 (see Section 3.1.5).

The energy levels of several redox couples as well as the positions of theband edges of typical semiconductors in equilibrium with aqueous elec-trolyte (pH 7) are compared in Fig. 5.59.

The electrochemical potential of an electron in a solid defines the Fermienergy (cf. Eq. 3.1.9). The Fermi energy of a semiconductor electrode (eF)and the electrolyte energy level (VREDOX) are generally different beforecontact of both phases (Fig. 5.60a). After immersing the semiconductorelectrode into the electrolyte, an equilibrium is attained:

— £REDOX (5.10.14)

The equilibration proceeds by electron transfer between the semiconductorand the electrolyte. The solution levels are almost intact ( £ R E DOX-£REDOX)> since the number of transferred electrons is negligible relative tothe number of the redox system molecules (cox and cred). On the other hand,the energy levels of the semiconductor phase may shift considerably. Theregion close to the interface is depleted of majority charge carriers and theenergy bands are bent upwards or downwards as depicted in Fig. 5.60b.

The equilibrium potential between the bulk of the semiconductor and thebulk of the electrolyte (Galvani potential, Af$) is expressed by Eq. (4.5.6).

Li7Li

LiVLi(Hg)

-4

-3

-2

I

H+/H2(pH7)Zn27Zn

SHE _ TQ

Fe37Fe2+ _ _ LO2/H2O(pH7)1

Ce /Ce

F2/F" _

"ii

e(eV)0

- -1

- -2

- -3

4

- - 5

- -6

- - 7

GaAs

GaP

CdS

Si

T iO 2

SrTiO,

Fig. 5.59 The energies of several redox couples (eTedox), and the lower edgeof the conduction band (ec) and the upper edge of the valence band (ev) oftypical semiconductors. The positions of band edges are given for semiconduc-

tors in equilibrium with aqueous electrolyte at pH 7

399

(a) (b) (c)

\

Fig. 5.60 The semiconductor/electrolyte interface (a) before equilibrationwith the electrolyte, (b) after equilibration with the electrolyte in the dark,and (c) after illumination. The upper part depicts the n-semiconductor and the

lower the p-semiconductor

This potential is related to the electrode potential, E (apart from an additiveconstant depending on the definition of the reference potential). The firstterm in Eq. (4.5.6), Af (j> characterizes the band bending

eAfcj) = esc - ec = el - ey (5.10.15)

where esc and ec are the energies of the lower edge of the conduction band at

the surface and in the bulk, respectively, and el and ey are analogousenergies of the upper edge of the valence band. The positions of es

c and elare determined by the Helmholtz electric layer, hence they remain fixed aslong as A|0 is not changed. External changes of the electrode potential, Ey

manifest themselves as changes of Afcf), at a certain electrode potential (theflatband potential, E^), the band bending disappears, Af0 = 0.

For a more detailed description of the semiconductor/electrolyte inter-face, it is convenient to define the quasi-Fermi levels of electrons, eFe andholes, eFp,

£FAX) = £c(x) + kT In [ne(x)/(Nc - ne(x))] (5.10.16)

and analogously

eFtP(x) = ey(x) - kT In [(np(x)/(Ny - np(x))] (5.10.17)

400

yvc and Nw are the effective densities of electronic states in the correspondingband edges which can be expressed as follows:

Nc = (2jtmekT/h2)3/2; Ny = (2jtmpkT/h2)3/2 (5.10.18)

me and mp are the effective masses of electron and hole, respectively; ne(x)and np(x) are the carrier concentrations defined by Eqs (4.5.1), (4.5.2) and(4.5.8). In equilibrium, both quasi-Fermi levels are equal:

eFtC(x) = eFtP(x) = eF (5.10.19)

for any x > 0. Combining Eqs (5.10.16)-(5.10.19) and assuming nc « Nc andftp«Nv, we can easily derive the equilibrium equations (4.5.8) and (4.5.9).

Upon illumination, photons having energy higher than the band gap(eg = ec — £v) are absorbed in the semiconductor phase and the electron-hole-pairs (e~/h+) are generated. This effect can be considered equivalentto the photoexcitation of a molecule (Fig. 5.57) if we formally identify theHOMO with the ec level and LUMO with the £v level. The lifetime ofexcited e~/h+ pairs (in the bulk semiconductor) is defined analogously asthe lifetime of the excited molecule in terms of a pseudo-first-orderrelaxation (Eq. 5.10.2).

The undesired recombination of the photogenerated e~/h+ pairs isconsiderably suppressed if the charge carriers are photogenerated in thespace charge region or in its close vicinity accessible by diffusion of excitedelectrons or holes during their lifetime. In this case, electrons and holes areseparated by the electric field of the space charge as depicted in Fig. 5.60c.The charge separation is apparently conditioned by the band bending, hencethe potential Af(f) must be more positive (n-semiconductors) or morenegative (p-semiconductors) than the flatband potential E^.

Under illumination, the electrons and holes within the space charge layerare no more in equilibrium defined by Eq. (5.10.19), i.e. the levels *eFe(jt)and *eFp(x) are different in this region. The concentration of the minoritycarriers near the surface is increased dramatically in comparison with theequilibrium value, hence the quasi-Fermi level of the minority carriersshows complicated dependence on x (Fig. 5.60c). On the other hand, thequasi-Fermi level of the majority charge carriers remains practically flat andis only shifted relative to eF. This shift occurs generally towards the initialvalue £p, i.e. the band bending decreases (Fig. 5.60c). The potential change(AZsph) corresponding to the shift of the bulk Fermi level *eF relative to eF

is called photopotential. It can simply be determined as a difference of thepotentials measured by using an ohmic contact at the backside of theilluminated (£LIGHT) and non-illuminated (£DARK) semiconductorelectrode:

AEph = {lie) |e? - eF| = |£L I G H T - ^DARKI (5.10.20)

The photopotential can also be expressed in terms of the relative change inthe minority carrier density, Anmin, introduced by photoexcitation. By using

Eqns (5.10.16), (5.10.17) and (5.10.20) we obtain

Anmjn°min)

401

(5.10.21)

The magnitude of the photopotential is also related to light intensity and tothe value of EDAKK. The value of EL1GHT cannot obviously exceed theflatband potential. At £DARK = E^, the photopotential drops to zero whichcan be used for a simple measurement of the flatband potential.

The photopontential also approaches to zero when the semiconductorphotoelectrode is short-circuited to a metal counterelectrode at which a fastreaction (injection of the majority carriers into the electrolyte) takes place.The corresponding photocurrent density is defined as a difference betweenthe current densities under illumination, /LIGHT and in the dark, /DARK-

7Ph — I/LIGHT /DARK I (5.10.22)

The photocurrent density (;ph) is proportional to the light intensity, butalmost independent of the electrode potential, provided that the bandbending is sufficiently large to prevent recombination. At potentials close tothe flatband potential, the photocurrent density again drops to zero. Atypical current density-voltage characteristics of an n-semiconductor elec-trode in the dark and upon illumination is shown in Fig. 5.61. If theelectrode reactions are slow, and/or if the e~/h+ recombination viaimpurities or surface states takes place, more complicated curves for/LIGHTresult.

The photocurrent is cathodic or anodic depending on the sign of theminority charge carriers injected from the semiconductor electrode into theelectrolyte, i.e. the n-semiconductor electrode behaves as a photoanode and

u

o1ft

^ph

^ j

.•max-/ph

—•~~~ - / L I G H T

^ ^ . -/DARK

E"E f t

Fig. 5.61 Current density-potential charac-teristics of n-semiconductor electrode in the

dark and upon illumination

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ANODE

Fig. 5.62 Scheme of a photovoltaic cell with n-semiconductor photoanode

the p-semiconductor electrode as a photocathode. The most commonphotovoltaic cells are based on the combination of a semiconductorphotoanode and a metal cathode (Fig. 5.62). Cells with a metal anode and asemiconductor photocathode or with a semiconductor photoanode and asemiconductor photocathode have, however, also been described.

The potential which controls the photoelectrochemical reaction is gener-ally not the photopotential defined by Eqs (5.10.20) and (5.10.21) (exceptfor the very special case where the values of *£v, £REDOX and the initialFermi energy of the counterelectrode are equal). The energy which drivesthe photoelectrochemical reaction, £R can be expressed, for example, for ann-semiconductor electrode as

eR = eg - - (sc - eF) (5.10.23)

This energy is consumed to overcome the overpotentials of the electrodereactions (rja, rjc), the IR drop in the external circuit and electrolyte, and itmight partly be converted into the free energy, AG, of stable products (ifany) of the endoergic electrochemical reaction:

£R = e(r?a + ??c + IR) + AG/NA (5.10.24)

The electrochemical cell can again be of the regenerative or electro-synthetic type, as with the photogalvanic cells described above. In theregenerative photovoltaic cell, the electron donor (D) and acceptor (A) (seeFig. 5.62) are two redox forms of one reversible redox couple, e.g.Fe(CN)2~/4~, I2/I", Br2/Br", S2~/Sx~, etc. the cell reaction is cyclic(AG = 0, cf. Eq. (5.10.24) since D+=A and D=A~). On the other hand, inthe electrosynthetic cell, the half-cell reactions are irreversible and theproducts (D+ and A~) accumulate in the electrolyte. The most carefullystudied reaction of this type is photoelectrolysis of water (D+ = O2 andy4~ = H2). Other photoelectrosynthetic studies include the preparation ofS2OI , the reduction of CO2 to formic acid, N2 to NH3, etc.

The photoelectroiysis of water was first studied in 1971 by A. Fujisima

403

and K. Honda, who used a n-TiO2 photoanode and a Pt counterelectrode.From the band positions and the formal energy levels of the H+/H2 andO2/H2O couples (Fig. 5.59), it is, nevertheless, apparent that the cell:

n-TiO2 |aqueous electrolyte| Pt (5.10.25)

does not provide sufficient voltage for photoelectrolysis of water (1.229 V)since the energy level of hydrogen electrode at pH 7 is about 0.2 V abovethe conduction band edge (this difference remains constant also at other pHvalues since both energy levels show the same Nernstian shift with pH). Themissing voltage has, therefore, to be supplied either from an external source(photoassisted electrolysis of water) or by immersing the electrodes intoelectrolytes of different pH values (photoanode in alkaline and cathode inacidic electrolyte, the electrolytes being separated by a suitable membrane).

The direct photoelectrolysis of water requires that the ey level be belowthe O2/H2O level and the ec level be above the H+/H2 level. This conditionis satisfied, e.g. for CdS, GaP, and several large-band gap semiconductors,such as SrTiO3, KTaO3, Nb2O5 and ZrO2 (cf. also Fig. 5.59). From thepractical points of view, these materials show, however, other specificproblems, e.g. low electrocatalytic activity, sensitivity to photocorrosion(CdS, GaP), and inconvenient absorption spectrum (oxides).

The world-wide effort in photo(electro)chemical splitting of water startedin the 1970s. It was motivated by two factors: (1) the oil crisis whichstimulated the research in the solar energy conversion, and (2) the optimismconnected with the 'hydrogen economy'. At present, however, the conver-sion of solar energy via photo(electro)lysis of water as well as the use ofhydrogen as a medium for the energy storage and transport do not seem tocompete seriously with the currently available energy resources.

5.10.5 Sensitization of semiconductor electrodes

The band-gap excitation of semiconductor electrodes brings two practicalproblems for photoelectrochemical solar energy conversion: (1) Most of theuseful semiconductors have relatively wide band gaps, hence they can beexcited only by ultraviolet radiation, whose proportion in the solar spectrumis rather low. (2) the photogenerated minority charge carriers in thesesemiconductors possess a high oxidative or reductive power to cause a rapidphotocorrosion.

Both these disadvantages can be overcome by sensitization. In this case,the light is not absorbed in the semiconductor phase, but in an organic ororganometallic photoredox active molecule (sensitizer, S), whose excitationenergy is lower than eg. The sensitizer might either be dissolved in theelectrolyte or adsorbed at the semiconductor surface. If the redox levels ofthe sensitizer in the ground and excited states are properly positionedrelative to the ec and £v levels, a charge injection from the photoexcitedsensitizer (S*) might occur. The photoexcited molecule reacts at the

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LOAD

~7~\

A"«-e-- A 0

Fig. 5.63 Scheme of a photoelectrochemical cell withsensitized semiconductor anode

interface to form the oxidized, S+ (reduced, S ) ground-state form via themajority charge carrier injection into the conduction (valence) band of then- (p-) type semiconductor electrode.

Practically more important is the sensitization of the n-type semiconduc-tor electrode (Fig. 5.63). The depicted scheme is virtually equivalent to thatin Fig. 5.62; the only exception is that the 'hole' is not created in the valenceband but formally in the sensitizer molecule.

The charge injection from the sensitizer, S*, dissolved in the electrolytesolution, might formally be considered as a photogalvanic process. The S*molecules must be able to diffuse to the semiconductor surface during theirlifetime, r0. Assuming the usual lifetime, T O ^ 1 0 ~ 6 to 10"4s and thediffusion coefficient, D of 10~5cm2/s, the effective diffusion distance of S*,(5D (see Eq. 2.5.9) is very small:

<5D = (jtDr)V2 — 56 nm (5.10.26)

Let us assume that the electrolyte is illuminated through the semiconduc-tor electrode as shown in Fig. 5.58 (this is, in principle, possible since thesemiconductor is transparent for wavelengths X>hcleg at which thesensitizer absorbs the radiation). The relative intensity of radiation trans-mitted to the distance <5D is given by the Lambert-Beer law:

log (///0) = -8cdD (5.10.27)

where e is the extinction coefficient of S and c its concentration. Thelight-harvesting efficiency 0 L is defined as:

<pL = (1 -1/Io) = 1 - 10~£C6D (5.10.28)

For a typical extinction coefficient e — 107cm2/mol (cf. Table 5.6) andc = 1 M, the calculated, (pL values (Eqs (5.9.42) and (5.9.44)) are only ca. 12per cent. The number of active S* molecules is further reduced by thequantum yield of the formation of the triplet state cpT (cf. Table 5.6) and thequantum yield of the injection of the majority charge carriers from S* into

405

the semiconductor electrode (<p,). Thus, the overall quantum yield ofphotocurrent (IPCE = incident photon to current efficiency) is a product ofall the three quantities:

IPCE = <pL • cpT • <p, (5.10.29)

It actually denotes the number of electrons flowing through the interface perone photon impinging on it, IPCE is usually considered for monochromaticlight (radiant power, P, in W/cm2) whose wavelength (A) matches thewavelength of the absorption maximum of the sensitizer, Amax. A morepractical equation for IPCE can easily be derived by combining thephotocurrent density, ;p h with the radiant power and wavelength:

IPCE =jphhc/XPe (5.10.30)

where c is the velocity of light.The light-harvesting efficiency of an electrode covered with an adsorbed

sensitizer can be expressed in the form similar to Eq. (5.10.28):

<pL=l-10-£ads r> (5.10.31)

where £ads is the extinction coefficient of adsorbed sensitizer, T is the surfaceconcentration of the adsorbate (in mol/cm2 of the physical surface area) andr is the roughness factor of the electrode. The latter is defined as the ratio ofphysical to geometrical surface areas (polycrystalline materials, such assemiconducting oxides may achieve the r values of about 102-103). Theextinction coefficient, eads, can be approximated by the usual extinctioncoefficient of the solution, e (in cm2/mol).

For a typical monomolecular coverage, T— 10"1() mol/cm2, an electroderoughness factor r = 1000 and an extinction coefficient eads= 107cm2/mol,the light-harvesting efficiency is, in comparison to the preceding case, veryhigh, cpL = 90 per cent. Moreover, the adsorbed sensitizer is in intimatecontact with the semiconductor surface, hence the conditions for chargeinjection from S* into the semiconductor are almost ideal (c^—>100 percent).

The most successful sensitizers so far tested are complexes of Ru(II) withvarious derivatives of 2,2' bipyridine, e.g. 2,2'-bipyridine 4,4'-dicarboxylicacid (L). The Ru(II)L3 complex is adsorbed from an aqueous solution ofsuitable pH value to oxidic semiconductors via electrostatic bonds between—COO~ groups of the ligands and the positively charged (protonized)semiconductor surface.

The sensitization of n-TiO2 by RuL3 was studied in detail by M. Gratzel etal. The photoelectrochemical reaction is initiated by an electron injectioninto the conduction band of TiO2:

*Ru(II)L3-> Ru(III)L3 + ec-b(TiO2) (5.10.32)

The standard redox potential of Ru(III)L3/Ru(II)L3 equals 1.56 V( proton-ated form). Even energetically highly demanding processes, such as

406

oxidation of water to oxygen (E{)= 1.229V) and bromide to Br2 (Eo =1.0873 V) are thermodynamically possible. Two reactions deserve specialattention, since they enable the construction of highly efficient regenera-tive cells

2Ru(III)U- + 21" -> 2Ru(II)U- 4-I2, IPCE (450 nm) - 70%

2Ru(III)U- + 2Br"-* 2Ru(II)L^" 4- Br2, IPCE (450 nm) - 55%

The best parameters of the I~/I2 cell were achieved by using anon-aqueous electrolyte and the anatase electrode of nanometer-sizedparticles, sensitized with Rul^ju - (NC)Ru(CN)(bpy)2)2. In 1991, B.O'Regan and M. Gratzel described a cell attaining parameters competitiveto commercial, solid-state photovoltaic devices.

5.10.6 Photoelectrochemical solar energy conversion

The solar spectrum corresponds roughly to the radiation of a black bodyat the temperature 5750 K. If we denote the photon flux in the wavelengthinterval from A to (A + dA) (in photons/m2 s) as /(A), the total solar powerflux (in W/m2) is given as:

Ps = hc\ /(A)dA (5.10.33)

The power flux corrected for atmospheric absorptions equals about1 kW/m2 if the Sun stands at zenith and the sky is clear. This situation iscustomarily labelled AM 1 (one standard air mass).

The total world consumption of energy (ca. 3.1020 J/year) can be met byconverting the solar energy impinging on about 0.01 per cent of the Earth'ssurface, which is about 0.3 per cent of the desert areas. The research onsolar energy conversion is further encouraged by the existence of naturalphotosynthesis, which provides all the fossile energy resources of the Earth.The natural photosynthesis transforms about 0.1 per cent of the solar energyimpinging on the Earth, but the energy conversion in green plants is not asefficient as that attained by currently available photovoltaic cells and otherartificial devices.

The solar energy conversion efficiency is defined as:

CE = Pout/Ps (5.10.34)

where Fout is the maximum power output of the cell per unit illuminatedarea. For a regenerative photoelectrochemical cell, the maximum value ofPout has to be derived from the photocurrent/potential characteristic of thecell (Fig. 5.64). The quality of the cell is also described by the so-called fillfactor,/:

J ~~ iout/V./ph ZA£-'ph ) \p.L\J.oJ)

where y™hax is the maximum photocurrent density (at short circuit) and

AZs™hax is the maximum photovoltage (at open circuit).

407

Jph

Fig. 5.64 Photocurrent density-voltagecharacteristics of the regenerative photo-electrochemical cell. The shaded area is the

maximum output power, Pout

Any quantum device, including photoelectrochemical cell, is in principlenot able to absorb (and convert) light whose wavelength is longer than acertain threshold, Ag. This wavelength corresponds to the photon energyequal to the excitation energy of the photoactive molecule (with photo-galvanic effects or sensitized semiconductors) or to the band gap of thesemiconductor photoelectrode:

he he(5.10.36)

The theoretically convertible fraction of the solar power equals:

A J odA (5.10.37)

The plot of CE = Pout/Fs (from Eqs (5.10.33) and (5.10.37)) versus Ag forAM 1.2 is shown in Fig. 5.65 (curve 1). It has a maximum of 47 per cent at1100 nm. Thermodynamic considerations, however, show that there areadditional energy losses following from the fact that the system is in athermal equilibrium with the surroundings and also with the radiation of ablack body at the same temperature. This causes partial re-emission of theabsorbed radiation (principle of detailed balance). If we take into accountthe equilibrium conditions and also the unavoidable entropy production, themaximum CE drops to 33 per cent at 840 nm (curve 2, Fig. 5.65).

Besides the mentioned thermodynamic losses, there always exist kineticlosses arising from the competitive non-radiative quenching of the excitedstate. For instance in photovoltaic devices, the undesired thermal recom-

408

CE (7o)

40

30

20

10

-

-

-

r

/ s

/

/

//

/

/ / X

/ / . ^ ^ - ^/ / / * * #s ^

/ / X S i >wI f / xI I / xI I / >/ / /

/ / y GaP

/

- 1-

-

* 2

-

\\

Ge

500 1000 1500

Xg (nm)

Fig. 5.65 Dependence of the solar conversion efficiency (CE)on the threshold wavelength (Ag) for a quantum converter atAM 1.2. Curve 1: Fraction of the total solar power convertibleby an ideal equilibrium converter with no thermodynamic andkinetic losses. Curve 2: As 1 but the inherent thermodynamiclosses (detailed balance and entropy production) are con-sidered. Continuous line: Efficiency of a regenerative photo-voltaic cell, where the thermodynamic and kinetic losses areconsidered. The values of Ag for some semiconductors are also

shown (according to J. R. Bolton et al.)

bination of photogenerated e /h+ pairs is prevented by a band bending.This leads to the necessary charge separation, but is also paid by additionalenergy losses (cf. Eq. 5.10.23), which can be estimated as about 0.6-0.8 eVper photon.

The theoretical solar conversion efficiency of a regenerative photovoltaiccell with a semiconductor photoelectrode therefore depends on the modelused to describe the thermodynamic and kinetic energy losses. The CEvalues, which consider all the mentioned losses can generally only beestimated; the full line in Fig. 5.65 represents such an approximation.Unfortunately, the materials possessing nearly the optimum absorptionproperties (Si, InP, and GaAs) are handicapped by their photocorrosionsensitivity and high price.

409

References

Aruchamy, A., see p. 342.Bard, A. J., R. Memming, and B. Miller, Terminology in semiconductor electro-

chemistry and photoelectrochemical energy conversion, Pure Appl. Chem., 63,569 (1991).

Brodsky, A. M., and Yu. V. Pleskov, Electron photoemission at a metal-electrolytesolution interface, Progr. Surf. ScL, 2, 1 (1972).

Cardon, F., W. P. Gomes, and W. Dekeyser (Eds), Photovoltaic and Photoelec-trochemical Solar Energy Conversion, Plenum Press, New York, 1981.

Connolly, J. S., Photochemical Conversion and Storage of Solar Energy, AcademicPress, New York, 1981.

Conway, B. E., Solvated Electrons in Field- and Photo-Assisted Processes atElectrodes, MAE 7, 83 (1972).

Finklea, H. O. see p. 343.Fox, M. A., and M. Chanon, Photoinduced Electron Transfer, Elsevier, Amsterdam

(1988).Gerischer, H., The impact of semiconductors on the concepts of electrochemistry,

Electrochim. Acta, 35, 1677 (1990).Gerischer, H., and J. Katz (Eds), Light-Induced Charge Separation in Biology and

Chemistry, Verlag Chemie, Weinheim, 1979.Gratzel, M., Heterogeneous Photochemical Electron Transfer Reactions, CRDC

Press, Baton Rouge, Florida, USA (1987).Hautala, R. R., R. B. King, and C. Kutel, Solar Energy, The Humana Press,

Clifton, 1979.Memming, R., Charge transfer processes at semiconductor electrodes, in

Electroanalytical Chemistry, Vol. 11, p. 1 (Ed. A. J. Bard), M. Dekker, NewYork, 1979; CTE, 7, 529 (1983).

Nozik, H. J., Photoelectrochemistry: applications to solar energy conversion, Ann.Rev. Phys. Chem., 29, 184 (1978).

O'Regan, B., and M. Gratzel, Nature, 353, 737 (1991).Pelizetti, E., and N. Serponne (Eds) Homogeneous and Heterogeneous

Photocatalysis, D. Reidel, Dordrecht, 1986.Photoelectrochemistry. Faraday Discussions of the Royal Society of Chemistry No.

70 (1980).Pleskov, Yu. V., Solar Energy Conversion. A Photoelectrochemical Approach,

Springer-Verlag, Berlin, 1990.Pleskov, Yu. V., and Yu. Ya. Gurevich, Semiconductor Photoelectrochemistry,

Plenum Press, New York, 1986.Santhanam, K. S. V., and M. Sharon, see p. 344.Sciavello, M. (Ed.), Photochemistry, Photocatalysis and Photoreactors, NATO ASI

Series C, Vol. 146, D. Reidel, Dordrecht (1985).Vlachopoulos, N., P. Liska, J. Augustynski, and M. Gratzel, J. Am. Chem. Soc,

110, 1216 (1988).

Chapter 6

Membrane Electrochemistry andBioelectrochemistry

Not only electric currents in muscles and nerves but particularly even themysterious effects occurring with electric fish can be explained by means of theproperties of semipermeable membranes.

W. Ostwald, 1891

6.1 Basic Concepts and Definitions

The discovery of galvanic electricity (i.e. electrical phenomena connectedwith the passage of electric current) by L. Galvani in 1786 occurredsimultaneously with his study of a bioelectrochemical phenomenon whichwas the response of excitable tissue to an electric impulse. E. duBois-Reymond found in 1849 that such electrical phenomena occur at thesurface of the tissue, but it was not until almost half a century later that W.Ostwald demonstrated that the site of these processes are electrochemicalsemipermeable membranes. In the next decade, research on semipermeablemembranes progressed in two directions—in the search for models ofbiological membranes and in the study of actual biological membranes.

The search for models of biological membranes led to the formation ofa separate branch of electrochemistry, i.e. membrane electrochemistry.The most important results obtained in this field include the theory andapplication of ion-exchanger membranes and the discovery of ion-selectiveelectrodes (including glass electrodes) and bilayer lipid membranes.

The study of biological membranes led to the conclusion that the greatmajority of the processes in biological systems occur at cell and organellemembranes. The electrochemical aspects of this subject form the basis ofbioelectrochemistry, dealing with the processes of charge separation andtransport in biological membranes and their models, including electron andproton transfer in cell respiration and photosynthesis as well as ion transportin the channels of excitable cells. The electrokinetic phenomena (electricaldouble layer, interfacial tension of cells and organelles, cell membraneextension and contraction, etc.) also belong to this field. Bioelectrochem-

410

411

istry includes the more classical subjects of the thermodynamics and kineticsof redox processes of components of biological systems at electrodes.

This chapter will deal with the basic properties of electrochemicalmembranes in general and the membrane aspects of bioelectrochemistry inparticular. A number of bioelectrochemical topics was discussed in Sections1.5.3 and 3.2.5.

6.1.1 Classification of membranesIn contrast to mechanics, where the term membrane (Lat. membrana =

parchment) designates an elastic, two-dimensional plate, this term is used inchemistry, biophysics and biology to designate a solid or liquid phase(usually, but not always, with a thickness substantially smaller than its otherdimensions) separating two, usually liquid, phases. The transport (permea-tion) of the various components of both phases through the membraneoccurs at different rates relative to those in the homogeneous phases withwhich the membrane is in contact. The membrane is consequently calledsemipermeable.

The thickness of the membrane phase can be either macroscopic('thick')—membranes with a thickness greater than micrometres—or micro-scopic ('thin'), i.e. with thicknesses comparable to molecular dimensions(biological membranes and their models, bilayer lipid films). Thick mem-branes are crystalline, glassy or liquid, while thin membranes possess theproperties of liquid crystals (fluid) or gels (crystalline).

Depending on their structure, membranes can be separated into porous,where matter is transported through pores in the membrane, and compact,where the substance is transported either through the entire homogeneousmembrane phase or its homogeneous parts.

Membranes can be homogeneous, where the whole membrane particip-ates in the permeation of a substance, or heterogeneous, where the activecomponent is anchored in a suitable support (for solid membranes) orabsorbed in a suitable diaphragm or acts as a plasticizer in a polymeric film.Both of the latter cases are connected with liquid membranes. Biologicalmembranes show heterogeneity at a molecular level.

Membranes exhibiting selectivity for ion permeation are termedelectrochemical membranes. These membranes must be distinguished fromsimple liquid junctions that are often formed in porous diaphragms (seeSection 2.5.3) where they only prevent mixing of the two solutions byconvection and have no effect on the mobility of the transported ions. It willbe seen in Sections 6.2 and 6.3 that the interior of some thick membraneshas properties analogous to those of liquid junctions, but that the mobilitiesof the transported ions are changed.

6.1.2 Membrane potentialsA characteristic property of electrochemical membranes is the formation

of an electric potential difference between the two sides of the membrane,

412

termed the membrane potential A$M:

d <H2)solution 1 | membrane | solution 2 (6.1.1)

x=p x=qSimilar to galvanic cells, the membrane potential is determined by subtract-ing the electric potential of the phase on the left from that of the phase onthe right, i.e.

A0M-A?0 = <K2)-<H1) (6.1.2)

For cell membranes, the intracellular liquid is usually denoted as solution 2,while solution 1 is the extracellular liquid.

The formation of a membrane potential is connected with the presence ofan electrical double layer at the surface of the membrane. For a thick,compact membrane, an electrical double layer is formed at both interfaces.The electrical double layer at a porous membrane is formed primarily in themembrane pores (see Section 6.2). The electrical double layer at thinmembranes is formed on both membrane surfaces. It is formed by fixed ionson the surface of the membrane and the diffuse layer in the electrolyte.

Consider the simple case where both sides of the membrane are in contactwith a solution of symmetrical electrolyte BA in a single solvent and themembrane is permeable for only one ionic species. In equilibrium itselectrochemical potential (Eq. (3.1.5)) in both solutions adjacent to themembrane has the same value. Thus,

if the membrane is permeable for cation Bz+ and

if the membrane is permeable for anion Az~.The membrane potential expressed by Eqs (6.1.3) and (6.1.4) is termed

the Nernst membrane potential as it originates from the analogous ideas asthe Nernst equation of the electrode potential (p. 165) and the equation ofthe Nernst potential at ITIES (Eq. (3.3.50)).

Consider a system in which both solutions contain various ions for whichthe membrane is permeable (diffusible ions) and one type of ion that, forsome reason (e.g. a macromolecular ion for a porous membrane), cannotpass through the membrane (non-diffusible ion). The membrane is perme-able for the solvent.

The equilibrium conditions for the diffusible ions are

fc(l,Pi) = £/(2,p2) (6.1.5)

413

This condition expresses the fact that the two solutions are under differentpressures, px and p2, as a result of their, in general, different osmoticpressures. An analogous equation cannot be written for the non-diffusibleion as it cannot pass through the membrane and the 'equilibrium'concentrations cannot be established.

First, let us consider dilute solutions, where it is possible to set px = p2-Then the electrochemical potentials in Eq. (6.1.5) are expanded in the usualmanner, yielding

^ 0 (6.1.6)

for the diffusible cation anda (2)^ O (6.1.7)

for the diffusible anion. Elimination of the terms containing the electricpotentials in these equations yields

(6.1.8)

i.e. for univalent, divalent, etc., cations and anions11/2

a+{2) \a2+{2)V12 ^o+( l ) La2+(1)J La3+(1)J

(6.1.9)= - = ALa2_(2)J La3_(2).

The constant A is termed the Donnan distribution coefficient.In the simple case of a diffusible, univalent cation B+ and anion A~ and

non-diffusible anion X~ present in phase 2, the condition of electroneutral-ity gives

For dilute solutions, activities can be replaced by concentrations in Eqs.(6.1.9), yielding

cB+(l)cA-(l) = cB+(2)cA-(2) (6.1.11)

Equations (6.1.10) and (6.1.11) give

A more exact solution of the equilibrium conditions (Eq. 6.1.5) mustconsider that the standard term /J° depends on the pressure, which isdifferent in the two solutions:

li°(T,p) = ii*(T,p)= fvdpJo

(6.1.13)

414

where /i* is the limiting value of JU° at /?—»0 and v is the molar volume ofthe component at pressure p. If the volume v is considered to beindependent of the pressure (more accurate calculations employ a lineardependence) then

li°(T,p) = iA*(T) + vp (6.1.14)

It can be readily seen that this procedure yields an equation whose left-handside is the same as that for Eqs (6.1.6) and (6.1.7) and whose right-handside is not equal to zero, but rather to v+(p\ — p2) or V-(pi — p2)-

Equations (6.1.6) and (6.1.7) yield the Donnan potential A0D = A0M inthe form

RTA0 D = ~ — ln A (6.1.15)

The Donnan potentials contain the individual ionic activities and cannot bemeasured by using a purely thermodynamic procedure. In the concentrationrange where the Debye-Hiickel limiting law is valid, the ionic activities canbe replaced by the mean activities.

The membrane potentials are measured by constructing a cell with asemipermeable membrane separating solutions 1 and 2:

Ag | AgCl, sat. KC11 solution 1 | solution 2 | sat. KC1, AgCl | Ag (6.1.16)

whose EMF is given by the sum of the membrane potential and the twoliquid junction potentials, which can usually be neglected.

Equations (6.1.3) and (6.1.4) could be derived from the relationships forthe liquid junction potential formed by a single salt (Eq. 2.5.31), assumingthat the anion mobility in Eq. (6.1.3) and the cation mobility in Eq. (6.1.4)equal zero. The general case, i.e. that the membrane only changes thetransport number of the cation and anion, was solved by J. Bernstein at thebeginning of this century using a relationship analogous to Eq. (2.5.31),which assumes the following form for a uni-univalent electrolyte:

, = (*_-*+) — l n - ^ (6.1.17)

where T+ and T_ are the transport numbers of the cation and anion in themembrane, respectively, which are different from the corresponding valuesin solution.

References

Blank, M. (Ed.), Bioelectrochemistry: Ions, Surfaces, Membranes, AmericanChemical Society, Washington, 1980.

Bioelectrochemistry, CTE, 10 (1985).Butterfield, D. A., Biological and Synthetic Membranes, John Wiley & Sons, New

York, 1989.

415

Eisenman, G. (Ed.), Membranes—A Series of Advances, M. Dekker, New York,Vol. 1, 1972; Vol. 2, 1973; Vol. 3, 1975.

Galvani, L., De Viribus Electricitatis in Motu Musculari, Typographia InstitutiScientiarum, Bologna, 1791, Translation: Green, R. M., Commentary on theEffect of Electricity on Muscular Motion, p. 66, E. Licht, Cambridge, Mass., 1953.

Jain, M. K., Introduction to Biological Membranes, John Wiley & Sons, New York,1988.

Keynes, R. D., The generation of electricity by fishes, Endeavour, 15, 215 (1956).Koryta, J., Ions, Electrodes and Membranes, 2nd ed. John Wiley & Sons,

Chichester, 1991.Koryta, J., What is bioelectrochemistry?, Electrochim. Acta, 29, 1291 (1984).Membrane phenomena, Discussion Faraday Soc, 21, 1956.Milazzo, G., and M. Blank (Eds), Bioelectrochemistry III—Charge Separation

Across Biomembranes, Plenum, New York, 1988.Overbeck, J. T., The Donnan equilibrium, in Progress in Biophysics and Biophysical

Chemistry, Vol. 6, p. 57, Pergamon Press, London, 1965.Schlogl, R., Stofftransport durch Membranen, D. Steinkopff, Darmstadt, 1964.Sollner, K., The early developments of the electrochemistry of polymer membranes,

in Charged Gells and Membranes (Ed. E. Selegny), Vol. 1, Reidel, Dordrecht,1976.

Williams, R. J. P., A general approach to bioelectrics, in Charge and Field Effects inBiosystems (Eds M. J. Allen and P. N. R. Usherwood), Abacus Press, London,1984.

Wu, C. H., Electric fish and the discovery of animal electricity, Am. Scientist, 72,598 (1984).

6.2 Ion-exchanger Membranes

Ion-exchanger membranes with fixed ion-exchanger sites contain ionconductive polymers (ionomers) the properties of which have already beendescribed on p. 128. These membranes are either homogeneous, consistingonly of a polyelectrolyte that may be chemically bonded to an un-ionizedpolymer matrix, and heterogeneous, where the grains of polyelectrolyte areincorporated into an un-ionized polymer membrane. The electrochemicalbehaviour of these two groups does not differ substantially.

All ion-exchanger membranes with fixed ion-exchanger sites are porous toa certain degree (in contrast to liquid membranes and to membranes ofion-selective electrodes based on solid or glassy electrolytes, such as a singlecrystal of lanthanum fluoride).

6.2.1 Classification of porous membranes

Depending on the pore size, porous membranes can be divided into threegroups:

(a) Membranes with wide pores are simply diaphragms limiting flow anddiffusion of the solutions with which the membrane is in contact. Whenthe diaphragm contains cylindrical pores with identical radii r and adensity N per unit area, then, in the ideal case, the material flux of the

416

ith component of the solution through the diaphragm is given as

Jt = c,/v - N^jzDXRTy^i-y1 (6.2.1)

where c, is the concentration of the ith component, D, is its diffusioncoefficient, jUz is its electrochemical potential and 7V is the volume flux ofthe solution:

Jy = d ^ = Njtr^dy1 ^ (6.2.2)

where dh is the mechanical permeability (permeability per unit pressuregradient), rj is the viscosity of the solution, d is the thickness of themembrane and P is the hydrostatic pressure. Equation (6.2.1) followsdirectly from Eq. (2.3.23) and (2.5.23).

(b) Fine-pore membranes (r = 1-100 nm) selectively affect the character oftransport. These are often called semipermeable membranes—thepermeability of the membrane is different for different components ofthe solution as a result of the properties of the membrane itself (ratherthan as a result of different mobilities of the components of thesolution). The walls of the pores of ion-exchanger membranes areelectrically charged and contain an aqueous solution. The contents ofthe electrolyte components in the pores depends on the electric chargeon the pore walls and concentration of the electrolyte in contact withthe membrane. The electrical double layer formed inside the poresresults in the counterions (=gegenions, ions with opposite charge to thefixed ions on the membrane wall) having a greater concentration in thepores than ions with the same charge as the fixed ions (coions). Whenthe solution is dilute and the pores so narrow that the diffuse electricallayer has an effective thickness comparable with the pore radius, thenthe gegenions are present in a clear excess over the coions (see Fig.6.1). In the extreme case, the electrical diffuse layers completely fill thepores, which then contain only counterions and the membrane ispermeable only for these ions, the transport number of counterionbeing T, = 1. Such a membrane is termed permselective. On the otherhand, if the electrolyte is more concentrated and the pores wider, thenthe excess of the counterions over the coions eventually becomesnegligible.

(c) Microporous membranes have such small pore radii that mass istransported by an exchange process between the dissolved species andthe solvent particles. This structure is characteristic, for example, foramorphous polymer films below the glass transition temperature. Theporosity is a result of irregular coiling of segments of the polymerchains. These membranes are used to separate mixtures of gases andliquids (these are not electrochemical membranes) and to desalinate

417

Fig. 6.1 Distribution of cations and anions in pores of a cation-exchangermembrane depends on pore radius which decreases in the sequence A—»B—»C.

In case C the membrane becomes permselective. (According to K. Sollner)

water by hyperfiltration. They must be very thin for transport to occur atall and must simultaneously be mechanically strong.

6.2.2 The potential of ion -exchanger membranes

The distribution of electric potential across the membrane and thedependence of the membrane potential on the concentration of fixed ions inthe membrane and of the electrolyte in the solutions in contact with themembrane is described in the model of an ion-exchanger membrane workedout by T. Teorell, and K. H. Meyer and J. F. Sievers.

This theory will be demonstrated on a membrane with fixed univalentnegative charges, with a concentration in the membrane, cx. The pores ofthe membrane are filled with the same solvent as the solutions with whichthe membrane is in contact that contain the same uni-univalent electrolytewith concentrations cx and c2. Conditions at the membrane-solutioninterface are analogous to those described by the Donnan equilibriumtheory, where the fixed ion X~ acts as a non-diffusible ion. The Donnanpotentials A0 D 1 = 0P — 0(1) and A0D 2 = 0(2) — 0q are established at bothsurfaces of the membranes (x = p and x = q). A liquid junction potential,A0L = 0q — 0P, due to ion diffusion is formed within the membrane. Thus

The Donnan equilibrium condition (6.1.9) givesc

- , 2 = C2(6.2.4)

418

(For simplicity, the effect of the activity coefficients is neglected; it shouldbe recalled that the membrane is not completely impermeable for anions sothat neither c_ p nor c_ q is equal to zero.) These relationships and theelectroneutrality condition in the membrane

yield the expressions for the surface concentrations at the membrane:

(6.2.5)

When the expressions (6.2.5) are substituted into the Henderson equation(2.5.34) A0L is obtained. Both contributions A$D are calculated from theDonnan equation. From Eq. (6.2.3) we obtain, for the membrane potential,

RT U+-U( 6 1 6 )

Figure 6.2 schematically depicts the concentration distribution in thision-exchanger membrane.

In the limiting case, Eq. (6.2.6) is converted to the relationships derivedabove: if the membrane is completely impermeable for the anion, thenc_ p = c_ q = 0 and also IL = 0 and the membrane potential is given by theNernst term (RT/F) In (cjc2) (cf. Eq. 6.1.3). If, on the other hand, the

membrane

Fig. 6.2 Concentration distribution of cationsc+, anions c_ and fixed anions in a cation-exchanging membrane. The bathing solutionshave electrolyte concentrations cx and c2. (Ac-

cording to K. Sollner)

419

membrane is completely permeable for both the cation and the anion, thencx = 0 as well as c+>q = c_ q = c2 and c+p = c_ p = cx and A0M is given by theHenderson term alone, -(RT/F)(t+ -t_) In (cjc^ (cf. Eq. 2.5.31).

6.2.3 Transport through a fine-pore membrane

Transport processes in ion-exchanger membranes are important in mostapplications. This section will first deal with interaction of the material fluxthrough the membrane with the field of the diffuse double layer in its poresand also with mutual interaction of the fluxes of various thermodynamicquantities, treated by using the thermodynamics of irreversible processes(the phenomenological description of membrane processes in terms of thethermodynamics of irreversible processes is the principal application of thisfield in electrochemistry). First, for example, a cation-exchanger membranewill be considered, containing an excess of mobile cations (counterions)over anions (coions). On the transport of electric current, the cations give agreater electrical impulse to the solvent molecules than the anions, leadingto formation of an electroosmotic flux in the direction of movement of thecations in the electric field.

An electroosmotic flux is formed as a result of the effect of the electricfield in the direction normal to the pores in the membrane, d<f>/dx, (assumedto be constant) on the charge in the electric diffuse layer in the pore with acharge density p. The charges move in the direction of the x axis (i.e. in thedirection of the field), together with the whole solution with velocity v. Atsteady state

dd> 32v_ r p = r/ (6.2.7)ax ay

where r\ is the viscosity of the liquid in the pores. According to this equationthe force acting on the charges is compensated by internal friction of theliquid. The coordinate in the direction perpendicular to the pore surface isdenoted by y. The Poisson equation (Eq. 1.3.8) holds for the space charge

(6.2.8)=

e dyso that substitution into Eq. (6.2.7) gives

The electrical potential in the layer of the liquid moving only negligibly atthe surface of the pores is equal to the electrokinetic potential £(see page 242). At the middle of the pore, the electric potential and its

420

gradient d(f>/dy equal zero and also dv/dy =0. Integration of Eq. (6.2.9)yields the Helmholtz-Smoluchowski equation

dcbetv=^r~ (6.2.10)

For a simple model of the membrane it is assumed that N cylindrical poreswith radius r are located perpendicular to a membrane of thickness d. Theresistance of such a membrane is given by the relationship

d d<bd^ (6.2.11)

j

where K is the conductivity of the pores of the membrane and j is thecurrent density (referred to the unit surface area of the membrane). Thevolume flux of the liquid through the membrane (electroosmotic flux) thenis

etiJv = vjir2N = — (6.2.12)

KTJ

Because of the electroosmotic flux, a cation-exchanger membrane is lessresistant to cation flux than to anion flux.

Some of the elements of thermodynamics of irreversible processes weredescribed in Sections 2.1 and 2.3. Consider the system represented by nfluxes of thermodynamic quantities and n driving forces; it follows fromEqs (2.1.3) and (2.1.4) that \n{n +1) independent experiments areneeded for determination of all phenomenological coefficients (e.g. bygradual elimination of all the driving forces except one, by gradualelimination of all the fluxes except one, etc.). Suitable selection of thedriving forces restricted by relationship (2.3.4) leads to considerablesimplification in the determination of the phenomenological coefficients andthus to a complete description of the transport process.

The theory of transport in membranes is based on the thermodynamics ofirreversible processes (Sections 2.1 and 2.3). If n species are involved in thetransport, we have to consider n independent fluxes (including that of thesolvent), each of them characterized by n phenomenological coefficients.Owing to the Onsager reciprocity relations, however, there are onlyn(n + l)/2 independent phenomenological coefficients, which can be deter-mined by as many independent experiments. As usual in the theory oftransport in electrolyte solutions, the driving forces are gradients ofelectrochemical potentials (in dilute solutions gradients of concentrationsand of the electric potential can also be used).

Consider the system shown in Fig. 6.3. The ion-exchanger membraneseparates solutions of a single, completely dissociated, uni-univalentelectrolyte. Two pistons can be employed to form a pressure differencebetween the two compartments. The two electrodes W^ and W2 are

421

Fig. 6.3 A membrane system during flow of electric current.Ap —p2 ~P\> Acs = cs(2) — cs(l). S denotes the current source

connected to an external source S to produce a flow of electric currentthrough the membrane. Two usually identical reference electrodes areplaced in the two compartments (e.g. two saturated calomel electrodes) andare used to measure the membrane potential.

Since the ionic fluxes cannot be measured individually, it is preferable tointroduce the salt flux, besides solvent flux and charge flux (current density).The driving forces would then be the gradients or differences of thechemical potentials in media with different salt concentrations and differentpressures, multiplied by —1. These differences must be relatively small toremain within the framework of linear irreversible thermodynamics, so that

RTAcsvsAp

(6.2.13)

where cs is the salt concentration, cw is the solvent concentration, cs and cw

are the average salt and solvent concentrations between the two compart-ments, respectively, vs and uw are the molar volumes of the salt and thesolvent, respectively, and Ap is the pressure difference between the twocompartments, t

Even then, the formulation of the system of fluxes and driving forces isnot yet satisfactory, because of the quantity Acw. Thus, a new definition ofthe fluxes will be introduced, i.e. the volume flux of the solution, defined bythe equation

vsJs

t An approximate expression for AjU, follows from the relationship

(6.2.14)

422

The conjugate driving force is the pressure gradient Ap multiplied by —1.Further, the relative flux of the dissolved substance compared to the flux ofthe solvent, i.e. the 'exchange' flux / D , is defined by the relationship

Jo = Jf-Jf (6.2.15)

where / s and /w are the fluxes of the salt and the solvent, respectively. Theconjugate driving force is then the osmotic pressure gradient (multiplied by- 1 ) , since (cf. Eq. 1.1.17)

AJZ = 2RTACS (6.2.16)

The dissipation function is given as

O = -JyAp - /DAJT - ; A 0 < 0 (6.2.17)

The present example of membrane transport is described by the followingset of equations:

Jv = -LvvAp - LsnAn - Lw<t>A0

Jr> = -LnvAp - Ln7lAx - L^Acj) (6.2.18)

; = -L^Ap - L^An - L^Acf)

containing six phenomenological coefficients, which must be determined inindependent experiments.

The coefficient L^ can be determined from the mechanical permeabilityof the membrane (cf. Eq. 6.2.2) in the presence of a pressure difference andabsence of a solute concentration difference (AJT — 0) and of an electricpotential difference A<j> (electrodes Wx and W2 are connected in short, thatis A(f) = 0). The electrical conductivity of the membrane gM in the absenceof a pressure and solute concentration difference gives the coefficient L^:

y = - g M A 0 = - L ^ A 0 (6.2.19)

Of the remaining three cross phenomenological coefficients, Lvjr can befound from the volume flux during dialysis (Ap = 0 , A(j) = 0, becauseelectrodes Wx and W2 are short-circuited):

^ (6.2.20)

Equations (6.2.18) yield the electroosmotic flux (Ap = 0 , An = 0) in theform

J-f-j* (6.2.21)

The sign of the cross coefficient Lw<t> determines the direction of theelectroosmotic flux and of the cation flux. From Eq. (6.2.12) we have

£*«^' (6.2.22)

423

where, for a cation-exchanger membrane, £ > 0 and, for an anion-exchanger membrane, £ < 0 . Thus, the electroosmotic flux for an anion-exchanger membrane will go in the opposite direction.

When a change in pressure Ap (the electroosmotic pressure) stops theelectroosmotic flux (/v

= 0> AJT = 0), then the first of Eqs (6.2.18) gives

(6.2.23)

Figure 6.4 depicts the formation of the electroosmotic pressure.The last experiment involves measurement of the membrane potential (in

the last of Eqs (6.2.18), j = 0 and Ap = 0), for which it follows from Eq.(6.2.16) that

+ Acs/2cs

- Acs/2cs

(6.2.24)

Fig. 6.4 Electroosmoticpressure. Hydrostaticpressure difference Apcompensates the osmoticpressure difference be-tween the compartments1 and 1' and prevents thesolvent from flowingthrough the membrane 2

424

From Eqs (6.1.17) and (6.2.19) we then obtain

, _ (j+ - T_)gM

L

It will now be demonstrated how Eqs (6.2.18) assist in the orientation inthe transport system, e.g. in the determination of the process that is suitablefor separation and accumulation of the components, etc. According to Eqs(6.2.14) and (6.2.15) the solute flux with respect to the membrane, /s , isgiven by the expression

j = Yf'_ JD + cs/v (6.2.26)fwcw + vscs

As the denominator in the first term on the right-hand side is equal to 1, weobtain with respect to Eqs (6.2.18) (A/7 = 0, A0 = 0)

/ s = -(Ljrjri;wcw + LV7r)csAjr (6.2.27)

By definition, Lnjt > 0. The sign of / s is determined by the sign of the crosscoefficient Lsn and its absolute value. If LVJT<0, the volume flux of thesolvent occurs in the direction from more dilute to more concentratedsolutions (i.e. in the direction of the osmotic pressure gradient). If \Lyn\ issmaller than Ljrjrvwcw, then the solute flows in the direction of the drop ofconcentration (in the opposite direction to the concentration or osmoticpressure gradient). This case is termed congruent flux. In the opposite case(-Lvjr > Ljrjruwcw), non-congruent flux is involved. For Lvjr > 0, the solventflux goes in the same direction as the salt flux, i.e. in the direction of theconcentration gradient of the salt. This interesting phenomenon, termednegative osmosis, has not yet been utilized in practice.

Electrodialysis is a process for the separation of an electrolyte from thesolvent and is used, for example, in desalination. This process occurs in asystem with at least three compartments (in practice, a large number isoften used). The terminal compartments contain the electrodes and themiddle compartment is separated from the terminal compartments byion-exchanger membranes, of which one membrane (1) is preferentiallypermeable for the cations and the other one (2) for the anions. Such asituation occurs when the concentration of the electrolyte in the compart-ments is less than the concentration of bonded ionic groups in themembrane. During current flow in the direction from membrane 1 tomembrane 2, cations pass through membrane 1 in the same direction andanions pass through membrane 2 in the opposite direction. In order for theelectrolyte to be accumulated in the central compartment, i.e. betweenmembranes 1 and 2 (it is assumed for simplicity that a uni-univalentelectrolyte is involved), the relative flux of the cations with respect to theflux of the solvent, /D,+, and the relative flux of the anions with respect to

425

that of the solvent, /D,-> must be related by the expression

/D,+ = - V - > 0 (6.2.28)

When Ap=0 (there is no pressure gradient in the system), L7lJl = 0(diffusion of ions through the membrane can be neglected), and if theconductivities of the two membranes are identical, then in view of thesecond of Eqs (6.2.18) this condition can be expressed as

L^ f + = - L ^ _ > 0 (6.2.29)

where Ln<t>t+ and Ln^_ are the phenomenological coefficients appearing inEqs (6.2.18) adjusted for the fluxes of the individual ionic species.

References

Bretag, A. H. (Ed.), Membrane Permeability: Experiments and Models, TechsearchInc., Adelaide, 1983.

Flett, D. S. (Ed.), Ion Exchange Membranes, Ellis Horwood, Chichester, 1983.Kedem, O., and A. Katchalsky, Permeability of composite membranes I, II, III,

Trans. Faraday Soc, 59, 1918, 1931, 1941 (1963).Lacey, R. E., and S. Loeb, Industrial Processes with Membranes, Wiley-

Interscience, New York, 1972.Lakshminarayanaiah, N., Equations of Membrane Biophysics, Academic Press,

Orlando, 1984.Meares, P. (Ed.), Membrane Separation Processes, Elsevier, Amsterdam, 1976.Meares, P., J. F. Thain, and D. G. Dawson, Transport across ion-exchange

membranes: The frictional model of transport, in Membranes—A Series ofAdvances (Ed. G. Eisenman), Vol. 1, p. 55, M. Dekker, New York, 1972.

Schlogl, R., see page 415.Spiegler, K. S., Principles of Desalination, Academic Press, New York, 1969.Strathmann, M., Membrane separation processes, /. Membrane ScL, 9, 121 (1981).Woermann, D., Selektiver Stofftransport durch Membranen, Ber. Bunseges., 83,

1075 (1979).

6.3 Ion-selective Electrodes

Ion-selective electrodes are membrane systems used as potentiometricsensors for various ions. In contrast to ion-exchanger membranes, theycontain a compact (homogeneous or heterogeneous) membrane with eitherfixed (solid or glassy) or mobile (liquid) ion-exchanger sites.

6.3.1 Liquid-membrane ion -selective electrodes

Liquid membranes in this type of ion-selective electrodes are usuallyheterogeneous systems consisting of a plastic film (polyvinyl chloride, siliconrubber, etc.), whose matrix contains an ion-exchanger solution as aplasticizer (see Fig. 6.5).

426

Fig. 6.5 Visible-light microscope photomicrograph showing pores (dark circles) in apolyvinyl chloride matrix membrane incorporating the ion-exchanger solution.

(From G. H. Griffiths, G. J. Moody and D. R. Thomas)

In electrophysiology ion-selective microelectrodes are employed. Theseelectrodes, resembling micropipettes (see Fig. 3.8), consist of glasscapillaries drawn out to a point with a diameter of several micrometres,hydrophobized and filled with an ion-exchanger solution, forming themembrane in the ion-selective microelectrode.

The ion-selective electrode-test solution (analyte)-reference electrodesystem is shown in the scheme

1 m 2ref. el. 11 B ^ | BXAX \ BXA2 | ref. el. 2 (f>3.\)

The membrane phase m is a solution of hydrophobic anion A{~(ion-exchanger ion) and cation Bx

+ in an organic solvent that is immisciblewith water. Solution 1 (the test aqueous solution) contains the salt of cationBj + with the hydrophilic anion A2~. The Gibbs transfer energy of anionsAi~ and A2~ is such that transport of these anions into the second phase isnegligible. Solution 2 (the internal solution of the ion-selective electrode)contains the salt of cation B ^ with anion A2~ (or some other similarhydrophilic anion). The reference electrodes are identical and the liquidjunction potentials A0L(1) and A0L(2) will be neglected.

The EMF of cell (6.3.1) is given by the relationship

E = E2 + A0L(2) + A2m4> + A0 L (m) + ^?<t> + A0L(1) - Ex

V6.3.2)

427

when the diffusion potential inside of the membrane, A0L(m) is neglected.With respect to Eq. (3.2.48),

E=^ln^ (6.3.3)F flBl+(2)

As the concentration of the internal solution of the ion-selective electrodeis constant, this type of electrode indicates the cation activity in the sameway as a cation electrode (or as an anion electrode if the ion-exchanger ionis a hydrophobic cation).

Consider that the test solution 1 contains an additional cation B2+ that is

identically or less hydrophilic than cation B ^ . Then the exchange reaction

B ^ m ) + B2+(w)«±B1

+(w) + B2+(m) (6.3.4)

occurs. The equilibrium constant of this reaction is (cf. page 63)

(A /^«0,O—»W A ^ « 0 , O — > W \

A G ^ B + - AG t r B2+ \ //: 1 c\~^~RT ) * *

As the concentration of the ion-exchanger ion is constant in the membranephase, it holds that

cBl+(m) + cB2+(m) = cA)-(m) (6.3.6)

For the potential difference &?</) (cf. Eqs 3.2.48 and 6.3.6) we have

l n

F cBl+(m)yB]+(m)

F cB2+(m)yB2+(m)

RT n (\ \ 4- Fv= T l n yR.+cA.-(n----—>— ( 6 3 7 )

Similarly, A^0 is given as

m<t> = — In — - — (6.3.8)

t aB]+(2)

The EMF of cell (6.3.1) can be written as

E = EISB-ETef (6.3.9)where the 'potential of the ion-selective electrode, £I S E' is expressed bycollecting the variables related to test solution 1 in one term and thevariables related to the membrane, the internal solution of the ion-selectiveelectrode and the internal reference electrode in another constant term sothat in view of (6.3.2), (6.3.7) and (6.3.8)

F y B 2 ( )RT

= £O,,SE + -j In [«„,•(!) + A:^32+aB2+(l)] (6.3.10)

428

The quantity K^iB2+ is termed the selectivity coefficient for the determinandB ^ with respect to the interferent B2

+. Obviously, for aB l+( l )»

^BT+,B2+«B2

+(1)5 Eq. (6.3.9) is converted to Eq. (6.3.3), often expressed bystating that the potential of the ion-selective electrode depends on thelogarithm of the activity of the determinand ion with Nernstian slope. Asimilar dependence is obtained for the ion B2

+ when aB l+ ( l )«

K^i,B2+0B2+(1)« Equation (6.3.10) is termed the Nikolsky equation.

Among cations, potassium, acetylcholine, some cationic surfactants(where the ion-exchanger ion is the p-chlorotetraphenylborate or tetra-phenylborate), calcium (long-chain alkyl esters of phosphoric acid asion-exchanger ions), among anions, nitrate, perchlorate and tetrafluoro-borate (long-chain tetraalkylammonium cations in the membrane), etc., aredetermined with this type of ion-selective electrodes.

Especially sensitive and selective potassium and some other ion-selectiveelectrodes employ special complexing agents in their membranes, termedionophores (discussed in detail on page 445). These substances, which oftenhave cyclic structures, bind alkali metal ions and some other cations incomplexes with widely varying stability constants. The membrane of anion-selective electrode contains the salt of the determined cation with ahydrophobic anion (usually tetraphenylborate) and excess ionophore, sothat the cation is mostly bound in the complex in the membrane. It canreadily be demonstrated that the membrane potential obeys Eq. (6.3.3). Inthe presence of interferents, the selectivity coefficient is given approximatelyby the ratio of the stability constants of the complexes of the two ions withthe ionophore. For the determination of potassium ions in the presence ofinterfering sodium ions, where the ionophore is the cyclic depsipeptide,valinomycin, the selectivity coefficient is A °+?Na+ ~ 10~4, so that thiselectrode can be used to determine potassium ions in the presence of a104-fold excess of sodium ions.

6.3.2 Ion -selective electrodes with fixed ion -exchanger sites

This type of membrane consists of a water-insoluble solid or glassyelectrolyte. One ionic sort in this electrolyte is bound in the membranestructure, while the other, usually but not always the determinand ion, ismobile in the membrane (see Section 2.6). The theory of these ion-selectiveelectrodes will be explained using the glass electrode as an example; this isthe oldest and best known sensor in the whole field of ion-selectiveelectrodes.

The membrane of the glass electrode is blown on the end of a glass tube.This tube is filled with a solution with a constant pH (acetate buffer,hydrochloric acid) and a reference electrode is placed in this solution (silverchloride or calomel electrodes). During the measurement, this whole systemis immersed with another reference electrode into the test solution. Themembrane potential of the glass electrode, when the internal and analysed

429

solutions are not too alkaline, is given by the equation

The theory of the function of the glass electrode is based on the conceptof exchange reactions at the surface of the glass. The glass consists of a solidsilicate matrix in which the alkali metal cations are quite mobile. The glassmembrane is hydrated to a depth of about 100 nm at the surface of contactwith the solution and the alkali metal cations can be exchanged for othercations in the solution, especially hydrogen ions. For example, the followingreaction occurs at the surface of sodium glass:

Na+(glass) + H+(solution) +± H+(glass) + Na+(solution) (6.3.12)

characterized by an ion-exchange equilibrium constant (cf. Eq. (6.3.5)).These concepts were developed kinetically by M. Dole and thermo-

dynamically by B. P. Nikolsky. The last-named author deduced a relation-ship that is analogous to Eq. (6.3.10). It is approximated by Eq. (6.3.11) forpH 1-10 and corresponds well to deviations from this equation in thealkaline region. The potentials measured in the alkaline region are, ofcourse, lower than those corresponding to Eq. (6.3.11). The differencebetween the measured potential and that calculated from Eq. (6.3.11) istermed the alkaline or 'sodium' error of the glass electrode, and depends onthe type of glass and the cation in solution (see Fig. 6.6). This dependence isunderstandable on the basis of the concepts given above. The cation of theglass can be replaced by hydrogen or by some other cation that is of thesame size or smaller than the original cation. The smaller the cation in theglass, the fewer the ionic sorts other than hydrogen ion that can replace itand the greater the concentration of these ions must be in solution for themto enter the surface to a significant degree. The smallest alkaline error isthus exhibited by lithium glass electrodes. For a given type of glass, theerror is greatest in LiOH solutions, smaller in NaOH solutions, etc.

Glass for glass electrodes must have rather low resistance, small alkalineerror (so that the electrodes can be used in as wide a pH range as possible)and low solubility (so that the pH in the solution layer around the electrodeis not different from that of the analyte). These requirements are contradic-tory to a certain degree. For example, lithium glasses have a small alkalineerror but are rather soluble. Examples of glasses suitable for glasselectrodes are, for example, Corning 015 with a composition of 72% SiO2,6% CaO and 22% Na2O, and lithium glass with 72% SiO2, 6% CaO and22% Li2O.

Glasses containing aluminium oxide or oxides of other trivalent metalsexhibit high selectivity for the alkali metal ions, often well into the acid pHregion. A glass electrode with a glass composition of llmol.% Na2O,18 mol.% A12O3 and 71 mol.% SiO2, which is sensitive for sodium ions and

v5- 0 .4 -

pH

Fig. 6.6 pH-dependence of the potential of the Beckman general, purpose glass electrode (Li2O, BaO, SiO2).(According to R. G. Bates)

431

poorly sensitive for both hydrogen and potassium ions, has found a wideapplication. At pH 11 its selectivity constant for potassium ions with respectto sodium ions is K^+^K+ = 3.6 x 10~3.

The membranes of the other ion-selective electrodes can be eitherhomogeneous (a single crystal, a pressed polycrystalline pellet) or heteroge-neous, where the crystalline substance is incorporated in the matrix of asuitable polymer (e.g. silicon rubber or Teflon). The equation controllingthe potential is analogous to Eq. (6.3.9).

Silver halide electrodes (with properties similar to electrodes of thesecond kind) are made of AgCl, AgBr and Agl. These electrodes,containing also Ag2S, are used for the determination of Cl~, Br~, I~ andCN~ ions in various inorganic and biological materials.

The lanthanum fluoride electrode (discussed in Section 2.6) is used todetermine F~ ions in neutral and acid media. After the pH-glass electrode,this is the most important of this group of electrodes.

The silver sulphide electrode is the most reliable electrode of this kindand is used to determine S2~, Ag+ and Hg2+ ions.

Electrodes containing a mixture of divalent metal sulphides and Ag2S areused to determine Pb2+, Cu2+ and Cd2+.

6.3.3 Calibration of ion -selective electrodes

It has been emphasized repeatedly that the individual activity coefficientscannot be measured experimentally. However, these values are required fora number of purposes, e.g. for calibration of ion-selective electrodes. Thus,a conventional scale of ionic activities must be defined on the basis ofsuitably selected standards. In addition, this definition must be consistentwith the definition of the conventional activity scale for the oxonium ion,i.e. the definition of the practical pH scale. Similarly, the individual scalesfor the various ions must be mutually consistent, i.e. they must satisfy therelationship between the experimentally measurable mean activity of theelectrolyte and the defined activities of the cation and anion in view of Eq.(1.1.11). Thus, by using galvanic cells without transport, e.g. a sodium-ion-selective glass electrode and a Cl~-selective electrode in a NaCl solution, aseries of a±(NaCl) is obtained from which the individual ion activity aNa+is determined on the basis of the Bates-Guggenheim convention for acl

(page 37). Table 6.1 lists three such standard solutions, where pNa =-logaNa+, etc.

6.3.4 Biosensors and other composite systems

Devices based on the glass electrode can be used to determine certaingases present in gaseous or liquid phase. Such a gas probe consists of a glasselectrode covered by a thin film of a plastic material with very small pores,

432

Table 6.1

Electrolyte

NaCl

NaF

CaCl2

Conventional standardsG. Bates and

Molality(mol • kg"1)

0.0010.010.11.00.0010.010.10.000 3330.003 330.033 30.333

of ion ;activities.M. Alfenaar)

pNa

3.0152.0441.1080.1603.0152.0441.108

pCa

3.5302.6531.8831.105

(According to R.

pCl

3.0152.0441.1100.204

3.1912.2201.2860.381

pF

3.1052.0481.124

which are hydrophobic, so that the solution cannot penetrate the pores. Athin layer of an indifferent electrolyte is present between the surface of thefilm and the glass electrode and is in contact with a reference electrode. Thegas (ammonia, carbon dioxide etc.) permeates through the pores of the film,dissolves in the solution at the surface of the glass electrode and induces achange of pH. For the determination of the concentration of the gas acalibration plot is used.

The Clark oxygen sensor is based on a similar principle. It contains anamperometric Pt electrode indicating the concentration of oxygen permeat-ing through the pores of the membrane. In enzyme electrodes a hydrophilicpolymer layer contains an enzyme which transforms the determinand into asubstance that is sensed by a suitable electrode. There are potentiometricenzyme electrodes based mainly on glass electrodes as, for example, the ureaelectrode, but the most important is the amperometric glucose sensor. Itcontains /3-glucose oxidase immobilized in a polyacrylamide gel and theClark oxygen sensor. The enzyme reaction is

glucose

glucose + O2 + H2O > H2O2 + gluconic acidoxidase °

The decrease of oxygen reduction current measured with the Clark oxygensensor indicates the concentration of glucose.

Enzyme electrodes belong to the family of biosensors. These also includesystems with tissue sections or immobilized microorganism suspensionsplaying an analogous role as immobilized enzyme layers in enzyme elec-trodes. While the stability of enzyme electrode systems is the most difficultproblem connected with their practical application, this is still more truewith the bacteria and tissue electrodes.

433

References

Ammann, D., Ion-Selective Microelectrodes, Springer-Verlag, Berlin, 1986.Baucke, F. G. K., The glass electrode—Applied electrochemistry of glass surfaces,

J. Non-Crystall. Solids, 73, 215 (1985).Berman, H. J., and N. C. Hebert (Eds), Ion-Selective Microelectrodes, Plenum

Press, New York, 1974.Cammann, K., Working with Ion-Selective Electrodes, Springer-Verlag, Berlin,

1979.Durst, R. A. (Ed.), Ion-Selective Electrodes, National Bureau of Standards,

Washington, 1969.Eisenman, G. (Ed.), Glass Electrodes for Hydrogen and Other Cations, M. Dekker,

New York, 1967.Freiser, H. (Ed.), Analytical Application of Ion-Selective Electrodes, Plenum Press,

New York, Vol. 1, 1978; Vol. 2, 1980.Janata, J., The Principles of Chemical Sensors. Plenum Press, New York, 1989.Koryta, J. (Ed.), Medical and Biological Applications of Electrochemical Devices,

John Wiley & Sons, Chichester, 1980.Koryta, J., and K. Stulik, Ion-Selective Electrodes, 2nd ed., Cambridge University

Press, Cambridge, 1983.Morf, W. E., The Principles of Ion-Selective Electrodes and of Membrane Transport,

Akademiai Kiado, Budapest and Elsevier, Amsterdam, 1981.Scheller, F., and F. Schubert, Biosensors, Elsevier, Amsterdam, 1992.Sykova, E., P. Hnik, and L. Vyklicky (Eds), Ion-Selective Microelectrodes and

Their Use in Excitable Tissues, Plenum Press, New York, 1981.Thomas, R. C , Ion-Sensitive Intracellular Microelectrodes. How To Make and Use

Them, Academic Press, London, 1978.Turner, A. P. F., I. Karube and G. S. Wilson, Biosensors, Fundamentals and

Applications, Oxford University Press, Oxford, 1989.Zeuthen, I. (Ed.), The Application of Ion-Selective Microelectrodes, Elsevier,

Amsterdam, 1981.

6.4 Biological Membranes

All living organisms consist of cells, and the higher organisms have amulticellular structure. In addition, the individual cells contain specializedformations called organelles like the nucleus, the mitochondria, theendoplasmatic reticulum, etc. (see Fig. 6.7). A great many life functionsoccur at the surfaces of cells and organelles, such as conversion of energy,the perception of external stimuli and the transfer of information, the basicelements of locomotion and metabolism. The large internal' surface of theorganism at which these processes occur leads to their immense variety. Thesurfaces of cells and cellular organelles consist of biological membranes. Incontrast to those discussed in the previous text these membranes areincomparably thinner. Electron microscope studies show them to consist ofthree layers with a thickness of 7-15 nm (see Fig. 6.8). In 1925, E. Gorterand F. Grendel isolated phospholipids from red blood cells and spread themout as a monolayer on a water-air interface. The overall surface of thismonolayer was approximately twice as large as the surface of the red bloodcells. The authors concluded that the erythrocyte membrane consists of a

434

mitochondrionT^Z

cytoplasmaticmembrane

lysosome

approx.Fig. 6.7 A scheme of an animal cell

bimolecular layer (bilayer) of phospholipids, with their long alkyl chainsoriented inwards (Fig. 6.9). Electron microscope studies have revealed thatthe central layer corresponds to this largely hydrocarbon region. In fact,biological membranes are still more complex as will be discussed in Section6.4.2.

6.4.1 Composition of biological membranes

Biological membranes consist of lipids, proteins and also sugars, some-times mutually bonded in the form of lipoproteins, glycolipids and gly-coproteins. They are highly hydrated—water forms up to 25 per cent of thedry weight of the membrane. The content of the various protein and lipidcomponents varies with the type of biological membrane. Thus, in

435

Fig. 6.8 Electron photomicrograph of mouse kidney mitochondria. The structureof both the cytoplasmatic membrane (centre) and the mitochondrial membranes isvisible on the ultrathin section. Magnification 70,000x. (By courtesy of J. Ludvik)

436

Fig. 6.9 Characteristic structures of biological membranes.(A) The fluid mosaic model (S. J. Singer and G. L.Nicholson) where the phospholipid component is pre-dominant. (B) The mitochondrial membrane where the

proteins prevail over the phospholipids

myelin cell membranes (myelin forms the sheath of nerve fibres) the ratioprotein: lipid is 1:4 while in the inner membrane of a mitochondrion (cf.page 464) 10:3.

The lipid component consists primarily of phospholipids and cholesterol.The most important group of phospholipids are phosphoglycerides, basedon phosphatidic acid (where X = H), with the formula

O

O H2C—O—C—R

R—C—O—CH

where R and R' are alkyl or alkenyl groups with long chains. Thus, glycerolis esterified at two sites by higher fatty acids such as palmitic, stearic, oleic,linolic, etc., acids and phosphoric acid is bound to the remaining alcoholgroup. In phospholipids, the phosphoric acid is usually bound to nitrogen-

437

substituted ethanolamines. For example, the phospholipid lecithin is formedby combination with choline (X = —CH2CH2N

+(CH3)3). Other moleculescan also be bound, such as serin, inositol or glycerol. Some phospholipidscontain ceramide in place of glycerol:

CH(OH)CH=CH(CH2)12CH3

CHNHCOR

CH2OH

where R is a long-chain alkyl group. The ester of ceramide with phosphoricacid bound to choline is sphingomyelin. When the ceramide is bound to asugar (such as glucose or galactose) through a /3-glycosidic bond, cere-brosides are formed. The representation of various lipidic species stronglyvaries among biological membranes. Thus, the predominant component ofthe cytoplasmatic membrane of bacterium Bacillus subtilis is phosphatidylglycerol (78 wt.%) while the main components of the inner membrane fromrat liver mitochondrion are phosphatidylcholine = lecithin (40 wt.%) andphophatidylethanolamine = cephalin (35 wt.%).

Cholesterol contributes to greater ordering of the lipids and thusdecreases the fluidity of the membrane.

Proteins either strengthen the membrane structure (building proteins) orfulfil various transport or catalytic functions (functional proteins). They areoften only electrostatically bound to the membrane surface (extrinsicproteins) or are covalently bound to the lipoprotein complexes (intrinsic orintegral proteins). They are usually present in the form of an or-helix orrandom coil. Some integral proteins penetrate through the membrane (seeSection 6.4.2).

Saccharides constitute 1-8 per cent of the dry membrane weight inmammals, for example, while this content increases to up to 25 per cent inamoebae. They are arranged in heteropolysaccharide (glycoprotein orglycolipid) chains and are covalently bound to the proteins or lipids. Themain sugar components are L-fucose, galactose, manose and sometimesglucose, N-acetylgalactosamine and N-acetylglucosamine. Sialic acid is animportant membrane component:

CH2OH

CHOH

CHOH

OH

This substance often constitutes the terminal unit in heteropolysaccharidechains, and contributes greatly to the surface charge of the membrane.

438

6.4.2 The structure of biological membranes

Following the original simple concepts of Gorter and Grendel, a largenumber of membrane models have been developed over the subsequent halfa century; the two most contrasting are shown in Fig. 6.9.

The basic characteristic of the membrane structure is its asymmetry,reflected not only in variously arranged proteins, but also in the fact that,for example, the outside of cytoplasmatic (cellular) membranes containsuncharged lecithin-type phospholipids, while the polar heads of stronglycharged phospholipids are directed into the inside of the cell (into thecytosol).

Phospholipids, which are one of the main structural components of themembrane, are present primarily as bilayers, as shown by molecularspectroscopy, electron microscopy and membrane transport studies (seeSection 6.4.4). Phospholipid mobility in the membrane is limited. Rota-tional and vibrational motion is very rapid (the amplitude of the vibration ofthe alkyl chains increases with increasing distance from the polar head).Lateral diffusion is also fast (in the direction parallel to the membranesurface). In contrast, transport of the phospholipid from one side of themembrane to the other (flip-flop) is very slow. These properties are typicalfor the liquid-crystal type of membranes, characterized chiefly by orderingalong a single coordinate. When decreasing the temperature (passing thetransition or Kraft point, characteristic for various phospholipids), theliquid-crystalline bilayer is converted into the crystalline (gel) structure,where movement in the plane is impossible.

The cells of the higher plants, algae, fungi and higher bacteria have a cellwall in addition to the cell membrane, protecting them from mechanicaldamage (e.g. membrane rupture if the cell is in a hypotonic solution). Thecell wall of green plants is built from polysaccharides, such as celluloseor hemicelluloses (water-insoluble polysaccharides usually with branchedstructure), and occasionally from a small amount of glycoprotein(sugar-protein complex). Because of acid groups often present in the poresof the cell wall it can behave as an ion-exchanger membrane (Section 6.2).

Gram-positive bacteria (which stain blue in the procedure suggested byH. Ch. J. Gram in 1884) have their cell walls built of cross-linked polymersof amino acids and sugars (peptidoglycans). The actual surface of thebacterium is formed by teichoic acid (a polymer consisting of a glycerol-phosphate backbone with linked glucose molecules) which makes ithydrophilic and negatively charged. In this case the cell wall is a sort ofgiant macromolecule—a bag enclosing the whole cell.

The surface structure of gram-negative bacteria (these are not stained byGram's method and must be stained red with carbol fuchsin) is morediversified. It consists of an outer membrane whose main building unit is alipopolysaccharide together with phospholipids and proteins. The actual cell

439

wall is made of a peptidoglycan. A loosely bounded polysaccharide layertermed a glycocalyx is formed on the surface of some animal cells.

6.4.3 Experimental models of biological membranes

Phospholipids are amphiphilic substances; i.e. their molecules containboth hydrophilic and hydrophobic groups. Above a certain concentrationlevel, amphiphilic substances with one ionized or polar and one stronglyhydrophobic group (e.g. the dodecylsulphate or cetyltrimethylammoniumions) form micelles in solution; these are, as a rule, spherical structures withhydrophilic groups on the surface and the inside filled with the hydrophobicparts of the molecules (usually long alkyl chains directed radially into thecentre of the sphere). Amphiphilic substances with two hydrophobic groupshave a tendency to form bilayer films under suitable conditions, withhydrophobic chains facing one another. Various methods of preparation ofthese bilayer lipid membranes (BLMs) are demonstrated in Fig. 6.10.

If a dilute electrolyte solution is divided by a Teflon foil with a smallwindow to which a drop of a solution of a lipid in a suitable solvent (e.g.octane) is applied, the following phenomenon is observed (Fig. 6.10A). Thelayer of lipid solution gradually becomes thinner, interference rainbowbands appear on it, followed by black spots and finally the whole layerbecomes black. This process involves conversion of the lipid layer from amultimolecular thickness to form a bilayer lipid membrane. Similar effectswere observed on soap bubbles by R. Hooke in 1672 and I. Newton in 1702.Membrane thinning is a result of capillary forces and dissolving ofcomponents of the membrane in the aqueous solution with which it is incontact. The thick layer at the edge of the membrane is termed thePlateau-Gibbs boundary. Membrane blackening is a result of interferencebetween incident and reflected light. If the membrane thickness is much lessthan one quarter of the wavelength of the incident light, then the waves ofthe incident and reflected light interfere and the membrane appears blackagainst a dark background. The membrane thickness can be found from thereflectance of light with a low angle of incidence, from measurements of themembrane capacity and from electron micrographs (application of a metalcoating to the membrane can, however, lead to artefact formation). Thethickness of a BLM prepared from different materials lies in the rangebetween 4 and 13 nm.

A BLM can even be prepared from phospholipid monolayers at thewater-air interface (Fig. 6.10B) and often does not then contain un-favourable organic solvent impurities. An asymmetric BLM can even beprepared containing different phospholipids on the two sides of themembrane. A method used for preparation of tiny segments of biologicalmembranes (patch-clamp) is also applied to BLM preparation (Fig. 6.IOC).

440

Teflon septum

drop ol lipidhexane solution

Plateau—Gibbs boundary

Fig. 6.10 Methods of preparation of bilayerlipid membranes. (A) A Teflon septum with awindow of approximately 1 mm2 area dividesthe solution into two compartments (a). A dropof a lipid-hexane solution is placed on thewindow (b). By capillary forces the lipid layer isthinned and a bilayer (black in appearance) isformed (c) (P. Mueller, D. O. Rudin, H. TiTien and W. D. Wescot). (B) The septum witha window is being immersed into the solutionwith a lipid monolayer on its surface (a). Afterimmersion of the whole window a bilayer lipidmembrane is formed (b) (M. Montal and P.Mueller). (C) A drop of lipid-hexane solution isplaced at the orifice of a glass capillary (a). Byslight sucking a bubble-formed BLM is shaped(b) (U. Wilmsen, C. Methfessel, W. Hanke and

G. Boheim)

The conductivity of membranes that do not contain dissolved ionophoresor lipophilic ions is often affected by cracking and impurities. The value fora completely compact membrane under reproducible conditions excludingthese effects varies from 10~8 to 10~10 Q"1 • cm"2. The conductivity of thesesimple 'unmodified' membranes is probably statistical in nature (as a resultof thermal motion), due to stochastically formed pores filled with water foran instant and thus accessible for the electrolytes in the solution with whichthe membrane is in contact. Various active (natural or synthetic) substances

441

! Teflon septum

-—window

B

BLM

suction

drop of liquid/hexanesolution

Fig. 6.10 (cont'd)

are often introduced into a BLM that preferably consists of syntheticphospholipids to limit irreproducible effects resulting from the use of naturalmaterials. Sometimes definite segments of biological membranes (e.g.photosynthetic centres of thylakoids) are imbedded in a BLM. Thedescribed type of planar BLM can be used for electrochemical measure-ments (of the membrane potential or current-potential dependence) andradiometric measurements of the permeation of labelled molecules.

The second model of a biological membrane is the liposome (lipidvesicle), formed by dispersing a lipid in an aqueous solution by sonication.In this way, small liposomes with a single BLM are formed (Fig. 6.11), witha diameter of about 50 nm. Electrochemical measurements cannot becarried out directly on liposomes because of their small dimensions. Afteraddition of a lipid-soluble ion (such as the tetraphenylphosphonium ion) tothe bathing solution, however, its distribution between this solution and theliposome is measured, yielding the membrane potential according to Eq.

442

Fig. 6.11 Liposome

(6.1.3). A planar BLM cannot be investigated by means of the molecularspectroscopical methods because of the small amount of substance in anindividual BLM. This disadvantage is removed for liposomes as they canform quite concentrated suspensions. For example, in the application ofelectron spin resonance (ESR) a 'spin-labelled' phospholipid is incorporatedinto the liposome membrane; this substance can be a phospholipid with, forexample, a 2,2,6,6-tetramethylpiperidyl-Af-oxide (TEMPO) group:

containing an unpaired electron. The properties of the ESR signal of thissubstance yield information on its motion and location in the membrane.

Liposomes also have a number of practical applications and can be used,for example, to introduce various Pharmaceuticals into the organism andeven for the transfer of plasmids from one cell to another.

6.4.4 Membrane transport

Passive transport. Similar to the lipid model, the non-polar interior of abiological membrane represents a barrier to the transport of hydrophilicsubstances; this is a very important function in preventing unwelcomesubstances from entering the cell and essential substances from leaving thecell. This property is demonstrated on a BLM which is almost impermeablefor hydrophilic substances. Polar non-electrolytes that are hydrated in theaqueous solution can pass through the membrane, with a high activationenergy for this process, 60-80 kJ • mol"1. This activation is primarily a resultof the large value of the Gibbs energy of transfer, i.e. the difference insolvation energy for the substance during transfer from the aqueous phaseinto the non-polar interior of the membrane.

In contrast, hydrophobic molecules and ions pass quite readily throughthe cell or model membrane. This phenomenon is typical for hydrophobic

443

ions such as tetraphenylborate, dipicrylaminate, triiodide and tetraalkylam-monium ions with long alkyl chains, etc. The transfer of hydrophobic ionsthrough a BLM with applied external voltage consists of three steps:

(a) Adsorption of the ion on the BLM(b) Transfer across the energy barrier within the BLM(c) Desorption from the other side of the BLM

As the membrane has a surface charge leading to formation of a diffuseelectrical layer, the adsorption of ions on the BLM is affected by thepotential difference in the diffuse layer on both outer sides of the membrane02 (the term surface potential is often used for this value in biophysics).Figure 6.12 depicts the distribution of the electric potential in the membraneand its vicinity. It will be assumed that the concentration c of the transferredunivalent cation is identical on both sides of the membrane and thatadsorption obeys a linear isotherm. Its velocity on the p side of themembrane (see scheme 6.1.1) is then

^ ^ £r(P) + £r(q) (6.4.1)

where F(p) and F(q) are the surface concentrations of the adsorbed ion onthe membrane surfaces, /ca and kd are the rates of adsorption anddesorption, respectively, cs is the volume concentration of the ion at thesurface of the membrane given by Eq. (4.3.5) and k and k are the rateconstants for ion transfer. This process can be described by using therelationship of electrochemical kinetics (5.2.24) for a = \\

(6.4.2)

Fig. 6.12 Electrical potential distribution inthe BLM and in its surroundings

444

where A0 is the applied potential difference. At steady state (dr(p)/df= dT(q)/df = 0),

T(P) =

r ( q ) = -td^d-ri.Ti,;

The conductivity of the membrane at the equilibrium potential (A0 = O),equal to the reciprocal of the polarization resistance value (Eq. 5.2.31),follows from Eqs (6.4.2), (6.4.3) and (4.3.5):

kd(kd + k + k)

kacs(kd

Ffc\RT r \ RTl

where ft = kjkd is the adsorption coefficient and c the bulk concentration ofthe ion.

Equation (6.4.4) is valid when the coverage of the electrolyte-membraneinterface is small. At higher concentrations of transferred ion, the iontransfer is retarded by adsorption on the opposite interface, so that thedependence of Go on c is characterized by a curve with a maximum, as hasbeen demonstrated experimentally.

Under certain conditions, the transfer of various molecules across themembrane is relatively easy. The membrane must contain a suitable'transport mediator', and the process is then termed 'facilitated membranetransport'. Transport mediators permit the transported hydrophilic sub-stance to overcome the hydrophobic regions in the membrane. Forexample, the transport of glucose into the red blood cells has an activationenergy of only 16 kJ • mol"1—close to simple diffusion.

Either the transport mediators bind the transported substances into theirinterior in a manner preventing them from contact with the hydrophobicinterior of the membrane or they modify the interior of the membrane sothat it becomes accessible for the hydrophilic particles.

A number of transport mediators are transport proteins; in the absence ofan external energy supply, thermal motion leads to their conformationalchange or rotation so that the transported substance, bound at one side ofthe membrane, is transferred to the other side of the membrane. This typeof mediator has a limited number of sites for binding the transportedsubstance, so that an increase in the concentration of the latter leads tosaturation. Here, the transport process is characterized by specificity for agiven substance and inhibition by other transportable substances competingfor binding sites and also by various inhibitors. When the concentrations ofthe transported substance are identical on both sides of the membrane,

445

exchange transport occurs (analogous to the exchange current—see Section5.2.2). When, for example, an additional substance B is added to solution 1under these conditions, it is also transported and competes with substance Afor binding sites on the mediator; i.e. substance A is transported fromsolution 2 into solution 1 in the absence of any differences in itsconcentration.

A further type of mediator includes substances with a relatively lowmolecular weight that characteristically facilitate the transport of ions acrossbiological membranes and their models. These transport mechanisms can bedivided into four groups:

(a) Transport of a stable compound of the ion carrier (ionophore) with thetransported substance.

(b) Carrier relay. The bond between the transported particle and theionophore is weak so that it jumps from one associate with theionophore to another during transport across the membrane.

(c) An ion-selective channel. The mediator is incorporated in a transversalposition across the membrane and permits ion transport. Examples aresubstances with helical molecular structure, where the ion passesthrough the helix. The selectivity is connected with the ratio of the helixradius to that of the ion.

(d) A membrane pore that permits hydrodynamic flux through the mem-brane. No selectivity is involved here.

A number of substances have been discovered in the last thirty years witha macrocyclic structure (i.e. with ten or more ring members), polar ringinterior and non-polar exterior. These substances form complexes withunivalent (sometimes divalent) cations, especially with alkali metal ions,with a stability that is very dependent on the individual ionic sort. Theymediate transport of ions through the lipid membranes of cells and cellorganelles, whence the origin of the term ion-carrier (ionophore). Theyion-specifically uncouple oxidative phosphorylation in mitochondria, whichled to their discovery in the 1950s. This property is also connected with theirantibiotic action. Furthermore, they produce a membrane potential on boththin lipid and thick membranes.

These substances include primarily depsipeptides (compounds whosestructural units consist of alternating amino acid and ar-hydroxy acid units).Their best-known representative is the cyclic antibiotic, valinomycin, with a36-membered ring [L-Lac-L-Val-D-Hy-i-Valac-D-Val]3, which was isolatedfrom a culture of the microorganism, Streptomyces fulvissimus. Figure 6.13depicts the structure of free valinomycin and its complex with a potassiumion, the most important of the coordination compounds of valinomycin.

Complex formation between a metal ion and a macrocyclic ligandinvolves interaction between the ion, freed of its solvation shell, and dipolesinside the ligand cavity. The standard Gibbs energy for the formation of thecomplex, AG?v, is given by the difference between the standard Gibbs

446

Fig. 6.13 Valinomycin structure in(A) a non-polar solvent (the 'brace-let' structure) and (B) its potassiumcomplex. O = O, o = C, # = N (Ac-cording to Yu. A. Ovchinnikov, V. I.

Ivanov and M. M. Shkrob)

energy for ion transfer from the vapour phase into the interior of the ligandin solution, AG?_^V> and the standard Gibbs energy of ion solvation, AG° j :

AGJV = AGj_*v ~~ AGs,j (6.4.5)

As the quantities on the right-hand side are different functions of the ionradius, the Gibbs energy of complex formation also depends on this quantity,but not monotonously. The dependence of the Gibbs energy of solvation, e.g.for the alkali metal ions, is an increasing function of the ion radius, i.e. ionswith a larger radius are more weakly solvated. In contrast, the interactionsof the desolvated ions with the ligand cavity are approximately identical ifthe radius of the ion is less than that of the cavity. If, however, the ion

447

radius is comparable to or even larger than the energetically mostfavourable cavity radius, then the ligand structure is strained or conforma-tional changes must occur. Thus, the dependence of the stability constant ofthe complex (and also the rate constant for complex formation) is usually acurve with a maximum.

The following conditions must be fulfilled in the ion transport through anion-selective transmembrane channel:

(a) The exterior of the channel in contact with the membrane must belipophilic.

(b) The structure must have a low conformational energy so that conforma-tional changes connected with the presence of the ion occur readily.

(c) The channel must be sufficiently long to connect both sides of thehydrophobic region of the membrane.

(d) The ion binding must be weaker than for ionophores so that the ion canrapidly change its coordination structure and pass readily through thechannel.

The polarity of the interior of the channel, usually lower than in the case ofionophores, often prevents complete ion dehydration which results in adecrease in the ion selectivity of the channel and also in a more difficultpermeation of strongly hydrated ions as a result of their large radii (forexample Li+).

The conditions a-d are fulfilled, for example, by the pentadecapeptide,valingramicidine A (Fig. 6.14):

HC=OI

L-Val-Gly-L-Ala-D-Leu-L-Ala-D-Val-L-Val-D-Val

NH-L-Try-D-Leu-L-Try-D-Leu-L-Try-D-Leu-L-Try

(CH2)2

OHwhose two helices joined tail-to-tail form an ion-selective channel. Theselectivity of this channel for various ions is given by the series

H+ > NH4+ S Cs+ > Rb+ > K+ > Na+ > Li+

If a substance that can form a transmembrane channel exists in severalconformations with different dipole moments, and only one of these formsis permeable for ions, then this form can be 'favoured' by applying anelectric potential difference across the membrane. The conductivity of themembrane then suddenly increases. Such a dependence of the conductivityof the membrane on the membrane potential is characteristic for themembranes of excitable cells.

448

Fig. 6.14 A gramicidinchannel consisting of twohelical molecules in thehead-to-head position.(According to V. I.

Ivanov)

If a small amount of gramicidin A is dissolved in a BLM (this substance iscompletely insoluble in water) and the conductivity of the membrane ismeasured by a sensitive, fast instrument, the dependence depicted in Fig.6.15 is obtained. The conductivity exhibits step-like fluctuations, with aroughly identical height of individual steps. Each step apparently cor-responds to one channel in the BLM, open for only a short time interval(the opening and closing mechanism is not known) and permits transport ofmany ions across the membrane under the influence of the electric field; inthe case of the experiment shown in Fig. 6.15 it is about 107 Na+ per secondat 0.1 V imposed on the BLM. Analysis of the power spectrum of these

Fig. 6.15 Fluctuation of the conductivity of BLM in thepresence of gramicidin A. (According to D. A. Haydon and B.

S. Hladky)

449

stochastic events (cf. page 373) has shown that the rate-detemining reactionof channel formation is the bimolecular reaction of two gramicidin molecules.

A similar effect has been observed for alamethicin I and II, hemocyanin,antiamoebin I and other substances. Great interest in the behaviour of thesesubstances was aroused by the fact that they represent simple models for ionchannels in nerve cells.

Some substances form pores in the membrane that do not exhibit ionselectivity and permit flow of the solution through the membrane. Theseinclude the polyene antibiotics amphotericin B,

OH O OH OH OH OH OHOOC

HO y "OHNH2

nystatin and mycoheptin, forming pores in the membrane with a diameter of0.7-1.1 nm. Proteins also form pores, e.g. the protein from the seaanemone Stoichactis helianthus or colicin El.

In all these systems, the energy source is an electrochemical potentialgradient and transport occurs in the direction — grad /i, (i.e. in the directionof decreasing electrochemical potential). It is often stated in the literaturethat this spontaneous type of transport occurs in the direction of theelectrochemical potential gradient; this is an imprecise formulation.

Transport occurring in the absence of another source of transport energyis termed passive transport.

Active transport. The definition of active transport has been a subject ofdiscussion for a number of years. Here, active transport is defined as amembrane transport process with a source of energy other than theelectrochemical potential gradient of the transported substance. This sourceof energy can be either a metabolic reaction (primary active transport) or anelectrochemical potential gradient of a substance different from that whichis actively transported (secondary active transport).

A classical example of active transport is the transport of sodium ions infrog skin from the epithelium to the corium, i.e. into the body. Theprincipal ionic component in the organism of a frog, sodium ions, is notwashed out of its body during its life in water. That this phenomenon is aresult of the active transport of sodium ions is demonstrated by anexperiment in which the skin of the common green frog is fixed as a

450

membrane between two separate compartments containing a single el-ectrolyte solution (usually Ringer's solution, 0.115 MNaCl, 0.002 MKCI and0.0018 MCaCl2). In the absence of an electric current a potential differenceis formed between the electrolyte at the outside and at the inside of theskin, equal to about —100 mV, even though the compositions of the twosolutions are identical. This potential difference decreases to zero whenelectrodes on both sides of the membrane are short-circuited and a currentflows between them. If the compartment on the outer side of the skincontains radioactively labelled 22Na, then sodium ions are shown to betransported from the outer side to the inner side of the skin. Sodium iontransport in this direction occurs even when the electrolyte in contact withthe inside of the skin contains a higher concentration of sodium ions thanthat at the outside.

When the temperature of the solution is increased, then the current aswell as the sodium transport rate increase far more than would correspondto simple diffusion or migration. When substances inhibiting metabolicprocesses are added to the solution, e.g. cyanide or the glycoside, ouabain,

OHCH3

the current decreases. For example, in the presence of 10" 4 M ouabain, itdecreases to 5 per cent of its original value.

The rate of the active transport of sodium ion across frog skin dependsboth on the electrochemical potential difference between the two sides ofthis complex membrane (or, more exactly, membrane system) and also onthe affinity of the chemical reaction occurring in the membrane. Thiscombination of material flux, a vector, and 'chemical flux' (see Eq. 2.3.26),which is scalar in nature, is possible according to the Curie principle onlywhen the medium in which the chemical reaction occurs is not homogeneousbut anisotropic (i.e. has an oriented structure in the direction perpendicularto the surface of the membrane or, as is sometimes stated, has a vectorialcharacter).

It is assumed that the chloride ion is transported passively across themembrane. Using an approach similar to the formulation of Eqs (2.1.2),(2.3.26) and (2.5.23), relationships can be written for the material fluxes ofsodium and chloride ions, 7Na+ and Jcr (the driving force is considered to bethe electrochemical potential difference), and for the flux of the chemicalreaction, /ch:

Jcr = L22AfiCr (6.4.6)/ch = LrlAjUNa+ + LrrA

451

where A/2Na+ and A/2cr designate the electrochemical potential differencebetween the inner and outer sides of the skin and A is the affinity of thechemical reaction. As the material flux in the direction from inside tooutside is considered as positive, then the coefficients Lu and L22 will alsobe positive. On the other hand, the coefficients LXr and LrX will be negative.

When the electrolyte concentration is identical on both sides of themembrane and A0M = 0, then AjUNa+ = AJUC, = 0 and Eqs (6.4.6) yield theequation for the current density,

7 = F / N a + = ^ / c h = L1H4 (6.4.7)L,rr

The resulting current is thus positive and proportional to the rate of themetabolic reaction.

Several types of active transport (also called a pump) can be distin-guished. Uniport is a type of transport in which only one substance istransported across the membrane. An example is the transport of protonsacross the purple membrane of the halophilic bacterium Halobacteriumhalobium, where the energy for transport is provided by light activating thetransport protein bacteriorhodopsin. The calcium pump in the membrane ofthe rod-shaped cells of the retina is also light-activated; the active substanceis the related rhodopsin. On radiation, its prosthetic group, retinal, istransformed from the cis form to the trans form and the protein componentis protonated by protons from the cytoplasm. After several milliseconds,retinal is converted back to the cis form and the protons are released intothe surrounding medium. Another example of uniport is the functioning ofH+-ATPase, described in detail in Section 6.5.2.

Symport is the interrelated transport of two substances in the samedirection, with opposite electrochemical potential gradients. Here theenergy of the electrochemical potential gradient of one substance is used foractive transport of the other substance. An example is the combinedtransport of sodium ions (in the direction of decreasing electrochemicalpotential) and glucose or amino acid (along its concentration gradient)across the membranes of the cell of the intestinal epithelium. In thisprocess, the organism transfers the products of digestion from the intestineinto the bloodstream.

Antiport is the coupling of two opposite transport processes. Na+,K+-ATPase activated by magnesium ions is a typical transport protein, enablingtransport of sodium and potassium ions against their electrochemicalpotential gradients. This process employs the energy gained from therupture of the 'macroergic' polyphosphate bond of adenosinetriphosphate(ATP) during its conversion to adenosinediphosphate (ADP) which iscatalysed by the enzyme, ATPase. In the actual process, the asparagine unitof the enzyme is first phosphorylated with simultaneous conformationalchanges, connected with the transfer of sodium ions from the cell into theintercellular fluid. The second step involves dephosphorylation of the

452

r~60A

-50 A

- 3 0 A

- 0A

--40 A

Fig. 6.16 The model of Na+,K+-ATPase from sheep kidney accord-ing to E. Amler, A. Abbott and W. J. Ball. The shading indicatesvarious functional areas of the dimeric enzyme:

|:::1 top of yet unknown functionbinding site of the nucleotide (ATP or ADP)phosphorylation siteion transporting part (with the channel)ouabain binding site

protein with transfer of a potassium ion from the intercellular fluid into thecell. The overall process corresponds to the scheme

Nao+

ATP + ATPase

ONH\ II

CH—CH.C—O

* ATPase 4- ADP

NH O\ II

H—CH2—C-

O

OH

(where the subscripts designate: i, intracellular and o, extracellular).The active form of the enzyme is, in fact, a dimer situated in the

phospholipid region of the cytoplasmic membrane (Fig. 6.16) and the ATPmolecule is bound between both the macromolecules.

This important enzyme is responsible, for example, for the active

453

transport of sodium ions in frog skin (see page 450), maintains a concentra-tion of sodium ions inside most cells lower than in the intercellular liquid(see page 454), etc. The function of Na+,K+-ATPase is reversible as itcatalyses the synthesis of ATP under artificial conditions that do not occur in theorganism (at high concentrations of Na+ inside the cells, low concentrationsof K+ outside the cells and high ADP concentrations).

From the point of view of the stoichiometry of the transported ions duringactive transport, the electroneutral pump, where there is no net chargetransfer or change in the membrane potential, must be distinguished fromthe electrogenic pump connected with charge transfer.

Active transport is of basic importance for life processes. For example, itconsumes 30-40 per cent of the metabolic energy in the human body. Thenervous system, which constitutes only 2 per cent of the weight of theorganism, utilizes 20 per cent of the total amount of oxygen consumed inrespiration to produce energy for active transport.

References

Abrahamson, S. A., and I. Pascher (Eds), Structure of Biological Membrane,Plenum Press, New York, 1977.

Amler, E., A. Abbott and W. J. Ball, Structural dynamics and oligometricinteractions of Na+,K+-ATPase as monitored using fluorescence energy transfer,Biophys. J., 61,553(1992).

Bittar, E. E. (Ed.), Membrane Structure and Function, John Wiley & Sons, NewYork, Vol. 1, 1980; Vol. 2, 1981.

Bittar, E. E. (Ed.), Membranes and Ion Transport, John Wiley & Sons, New York,Vol. 1, 1970; Vol. 2, 1971; Vol. 3, 1971.

Bures, J., M. Petran, and J. Zachar, Electrophysiological Methods in BiologicalResearch, Academic Press, New York, 1967.

Ceve, G., and D. Marsh, Phospholipid Bilayers, John Wiley & Sons, New York,1987.

Chaplan, S. R., and A. Essig, Bioenergetics and Linear Nonequilibrium Therm-odynamics. The Steady State, Harvard University Press, Cambridge, Mass., 1983.

Chapman, D. (Ed.), Biological Membranes, a series of advances published since1968, Academic Press, London.

Chapman, D. (Ed.), Biomembrane Structure and Function, Vol. 4 of Topics inMolecular and Structural Biology, Verlag Chemie, Weinheim, 1984.

Eisenman, G., J. Sandblom, and E. Neher, Interactions in cation permeationthrough the gramicidin channel—Cs, Rb, K, Na, Li, Tl, H and effects of anionbinding, Biophys. J., 22, 307 (1978).

Fineau, J. B., R. Coleman, and R. H. Michell, Membranes and Their CellularFunction, 2nd ed., Blackwell, Oxford, 1978.

Gomez-Puyon, A., and C. Gomez-Lojero, The use of ionophores and channelformers in the study of the function of biological membranes, in Current Topics inBioenergetics, Vol. 6, p. 221, Academic Press, New York, 1977.

Gregoriadis, G. (Ed.), Liposome Technology, Vols 1-3, CRC Press, Boca Raton,1984.

Haydon, D. A., and B. S. Hladky, Ion transport across thin lipid membranes: acritical discussion of mechanisms in selected systems, Quart. Rev. Biophys., 5, 187(1972).

454

Henderson, R., The purple membrane from Halobacterium halobium, Ann. Rev.Biophys. Bioeng., 6, 87 (1977).

Katchalsky, A., and P. F. Curran, Non-Equilibrium Thermodynamics in Biophysics,Harvard University Press, Cambridge, Mass., 1967.

Knight, C. G. (Ed.), Liposomes: From Physical Structure to TherapeuticApplications, Elsevier, North-Holland, Amsterdam, 1981.

Kotyk, A., and K. Janacek, Cell Membrane Transport, Principles and Techniques,2nd ed., Plenum Press, New York, 1975.

Lauger, P., Transport noise in membranes, Biochim. Biophys. Acta, 507, 337(1978).

Martonosi, A. N. (Ed.), Membranes and Transport, Vols. 1 and 2, Plenum Press,New York, 1982.

Montal, M., and P. Mueller, Formation of bimolecular membranes from iipidmonolayers and a study of their electrical properties, Proc. Natl. Acad. Sci. USA,69, 3561 (1972).

Ovchinnikov, Yu. A., V. I. Ivanov, and M. M. Shkrob, Membrane ActiveComplexones, Elsevier, Amsterdam, 1974.

Papahadjopoulos, D. (Ed.), Liposomes and Their Use in Biology and Medicine,Annals of the New York Academy of Science, Vol. 308, New York, 1978.

Pullmann, A. (Ed.), Transport Through Membranes: Carriers, Channels andPumps, Kluwer, Dordrecht, 1988.

Singer, S. I., and G. L. Nicholson, The fluid mosaic model of the structure of cellmembranes, Science, 175, 720 (1977).

Skou, J. C , J. G. Norby, A. B. Maunsbach and M. Esmann (Eds), The Na+,K+-Pump, Part A, A. R. Liss, New York, 1988.

Spack, G. (Ed.), Physical Chemistry of Transmembrane Motions, Elsevier, Amster-dam, 1983.

Stryer, L., Biochemistry, W. H. Freeman & Co., New York, 1984.Ti Tien, H., Bilayer Lipid Membrane (BLM), M. Dekker, New York, 1974.Urry, D. W., Chemical basis of ion transport specificity in biological membranes, in

Topics in Current Chemistry (Ed. F. L. Boschke), Springer-Verlag, Berlin.Wilschiit, J., and D. Hoekstra, Membrane fusion: from liposomes to biological

membranes, Trends Biochem. ScL, 9, 479 (1984).

6.5 Examples of Biological Membrane Processes

6.5.1 Processes in the cells of excitable tissues

The transport of information from sensors to the central nervous systemand of instructions from the central nervous system to the various organsoccurs through electric impulses transported by nerve cells (see Fig. 6.17).These cells consist of a body with star-like projections and a long fibrous tailcalled an axon. While in some molluscs the whole membrane is in contactwith the intercellular liquid, in other animals it is covered with a multiplemyeline layer which is interrupted in definite segments (nodes of Ranvier).The Na+,K+-ATPase located in the membrane maintains marked ionicconcentration differences in the nerve cell and in the intercellular liquid. Forexample, the squid axon contains 0.05 MNa+, 0.4 M K + , 0.04-0.1 MC1~,0.27 M isethionate anion and 0.075 M aspartic acid anion, while the inter-cellular liquid contains 0.46 M Na+, 0.01 M K+ and 0.054 M Cl".

The relationship between the electrical excitation of the axon and themembrane potential was clarified by A. L. Hodgkin and A. P. Huxley in

455

oCQ

oX

Fig. 6.17 A scheme of motoric nerve cell

experiments on the giant squid axon with a thickness of up to 1 mm. Theexperimental arrangement is depicted in Fig. 6.18. The membrane potentialis measured by two identical reference electrodes, usually micropipettesshown in Fig. 3.8. If the axon is not excited, the membrane potentialA0M= $(in) - 0(out) has a rest value of about -90 mV. When the cell isexcited by small square wave current impulses, a change occurs in themembrane potential roughly proportional to the magnitude of the excitationcurrent impulses (see Fig. 6.19). If current flows from the interior of the cellto the exterior, then the absolute value of the membrane potential increasesand the membrane is hyperpolarized. Current flowing in the oppositedirection has a depolarization effect and the absolute potential value

456

oscilloscope

EXTRACELLULARLIQUID

AXON

Fig. 6.18 Experimental arrangement for measurement of the mem-brane potential of a nerve fibre (axon) excited by means of currentpulses: (1) excitation electrodes, (2) potential probes. (According to B

Katz)

decreases. When the depolarization impulse exceeds a certain 'threshold'value, the potential suddenly increases (Fig. 6.19, curve 2). The characteris-tic potential maximum is called a spike and its height no longer depends ona further increase in the excitation impulse. Sufficiently large excitation ofthe membrane results in a large increase in the membrane permeability forsodium ions so that, finally, the membrane potential almost acquires thevalue of the Nernst potential for sodium ions (A0M = +50 mV).

Acp,mV

t,ms

Fig. 6.19 Time dependence of excitation current pulses(1) and membrane potential A0. (2) The abrupt peak is

the spike. (According to B. Katz)

457

A potential drop to the rest value is accompanied by a temporary influx ofsodium ions from the intercellular liquid into the axon.

If the nerve is excited by a 'subthreshold' current impulse, then a changein the membrane potential is produced that disappears at a small distancefrom the excitation site (at most 2 mm). A spike produced by a threshold orlarger current impulse produces further excitation along the membrane,yielding further spikes that are propagated along the axon. As alreadypointed out, sodium ions are transferred from the intercellular liquid intothe axon during the spike. This gradual formation and disappearance ofpositive charges corresponds to the flow of positive electric current alongthe axon. An adequate conductance of thick bare cephalopod axons allowsthe flow of sufficiently strong currents. In myelinized axons of vertebrates amuch larger charge is formed (due to the much higher density of sodiumchannels in the nodes of Ranvier) which moves at high speed throughmuch thinner axons than those of cephalopods. The myelin sheath theninsulates the nerve fibre, impeding in this way the induction of an oppositecurrent in the intercellular liquid which would hinder current flow inside theaxon.

The electrophysiological method described above is, in fact, a galvano-static method (page 300). A more effective method (the so-called voltage-clamp method) is based on polarizing the membrane by using a four-electrode potentiostatic arrangement (page 295). In this way, Cole, Hodgkinand Huxley showed that individual currents linked to selective ion transferacross the membrane are responsible for impulse generation and propaga-tion. A typical current-time curve is shown in Fig. 6.20. Obviously, themembrane ion transfer is activated at the start, but after some time itbecomes gradually inhibited. The ion transfer rate typically depends on both

-18 mV

L -78 mV

10

iime (ms)

Fig. 6.20 Time dependence of the membrane current. Sincethe potassium channel is blocked the current corresponds tosodium transport. The upper line represents the time course ofthe imposed potential difference. (According to W. Ulbricht)

458

the outer (bathing) and inner solution (the inside of a cephalopodmembrane as much as 1 mm thick can be rinsed with an electrolyte solutionwithout affecting its activity). The assumption that the membrane currentsare due to ion transfer through ion-specific channels was shown correct bymeans of experiments where the channels responsible for transfer of acertain ionic species were blocked by specific agents. Thus, the sodiumtransfer is inhibited by the toxin, tetrodotoxin:

H

found in the gonads and in the liver of fish from the family Tetraodontidae(e.g. the Japanese delicacy, fugu) and by a number of similar substances.Their physiological effect is paralysis of the respiratory function. Thetransport of potassium ions is blocked by the tetraalkylammonium ion withthree ethyl groups and one longer alkyl group, such as a nonyl.

The effect of toxins on ion transport across the axon membrane, whichoccurs at very low concentrations, has led to the conclusion that themembrane contains ion-selective channels responsible for ion transport.This assumption was confirmed by analysis of the noise level inionic currents resulting from channel opening and closing (cf. page 448). Single-channel recording was a decisive experiment. Here, a glass capillary (amicropipette) with an opening of about 1 jum2 is pressed onto the surface ofthe cell and a very tight seal between the phospholipids of the membraneand the glass of the micropipette is formed (cf. page 441). Because of thevery low extraneous noise level, this patch-clamp method permits themeasurement of picoampere currents in the millisecond range (Fig. 6.21).Obviously, this course of events is formally similar to that observed with thegramicidin channel (page 448) but the mechanism of opening and closing isdifferent. In contrast with the gramicidin channel, the nerve cell channelsare much larger and stable formations. Usually they are glycoproteins,consisting of several subunits. Their hydrophobic region is situated insidethe membrane while the sugar units stretch out. Ion channels of excitablecells consist of a narrow pore, of a gate that opens and closes the access tothe pore, and of a sensor that reacts to the stimuli from outside and issuesinstructions to the gate. The outer stimuli are either a potential change orbinding of a specific compound to the sensor.

The nerve axon sodium channel was studied in detail (in fact, as shown bythe power spectrum analysis, there are two sorts of this channel: one withfast opening and slow inactivation and the other with opposite properties).It is a glycoprotein consisting of three subunits (Fig. 6.22), the largest (mol.wt. 3.5 x 105) with a pore inside and two smaller ones (mol.wt 3.5 x 104 and3.3 x 104). The attenuation in the orifice of the pore is a kind of a filter

10

20

30

I 400)

| „ 50

PE E 60

70

80-

JL

1

nJUl fUUUL

n n

o— c

o

Ifnn JL

co

lnmnnnni:

5pA

n n n n n n n n c io §

time

Fig. 6.21 Joint application of patch-clamp and voltage-clamp methods to the study of a single potassium channelpresent in the membrane of a spinal-cord neuron cultivatedin the tissue culture. The values indicated before each curveare potential differences imposed on the membrane. The ionchannel is either closed (C) or open (O). (A simplified

drawing according to B. Hille)

polysaccharide

selective filter

Fig. 6.22 A function modelof the sodium channel. P den-otes protein, S the potentialsensitive sensor and H thegate. The negative sign marksthe carboxylate group wherethe guanidine group of tetro-dotoxin can be attached. (Ac-

cording to B. Hille)

460

(0.5 x 0.4 nm in size) controlling the entrance of ions with a definite radius.The rate of transport of sodium ions through the channel is considerable:when polarizing the membrane with a potential difference 4- 60 mV acurrent of approximately 1.5 pA flows through the channel which cor-responds to 6 x 106 Na+ ions per second—practically the same value as withthe gramicidin A channel. The sodium channel is only selective but notspecific for sodium transport. It shows approximately the same permeabilityto lithium ions, whereas it is roughly ten times lower than for potassium.The density of sodium channels varies among different animals, being only30 fim~2 in the case of some marine animals and 330 jurn"2 in the squidaxon, reaching 1.2 x 104 fj,m~2 in the mammalian nodes of Ranvier (see Fig.6.17).

The potassium channel mentioned above (there are many kinds) is morespecific for K+ than the sodium channel for Na+ being almost impermeableto Na+.

200 msec

-120 mV

-80 mV

KXJT ^ j A ^ J A n n r ^ ^

-40 mV

Fig. 6.23 Single-channel currents flowing across the membrane between theprotoplast and vacuole of Chara corallina. Among several channels withdifferent conductivity the recordings of the 130 pS channel are recorded here.

The zero line is at the top of each curve. (By courtesy of F. Homble)

461

The stochastic nature of membrane phenomena originating in channelopening and closing is not restricted to excitable cells. Figure 6.23 shows thetime dependence of currents flowing through a patch of the membranebetween the tonoplast (protoplasma) and the vacuole in the isolated part ofa cell of the freshwater alga, Char a corallina. In this membrane there arethree types of potassium channels with different conductivity and thebehaviour of the 130 pS channel is displayed in the figure.

D. E. Goldman, A. L. Hodgkin, A. P. Huxley and B. Katz (A. L. H.and A. P. H., Nobel Prize for Physiology and Medicine, 1963, B.K., 1970)developed a theory of the resting potential of axon membranes, based onthe assumption that the strength of the electric field in a thin membrane isconstant and that ion transport in the membrane can be described by theNernst-Planck equations. It would appear that this approach does notcorrespond to reality—it has been pointed out that ions are transportedthrough the membrane in channels that are specific for a certain kind of ion.Thus, diffusion is not involved, but rather a jumping of the ions through themembrane, that must overcome a certain energy barrier.

In deriving a relationship for the resting potential of the axon membraneit will be assumed that, in the vicinity of the resting potential, the frequencyof opening of a definite kind of ion channel is not markedly dependent onthe membrane potential. The transport of ions through the membrane canbe described by the same equations as the rate of an electrode reaction inSection 5.2.2. It will be assumed that the resting potential is determined bythe transport of potassium, sodium and chloride ions alone. The constants fcpare functions of the frequency of opening and closing of the gates of theion-selective channels. The solution to this problem will be based onanalogous assumptions to those employed for the mixed potential (seeSection 5.8.4).

The material fluxes of the individual ions are given by the equations

exp (=%£*) - *<r+CK+(2) exp

•W = *2T.*cNa+(l) exp ( ^ y M ) - *lf.*CN.*(2) exp (^f) (6-5.

Jcr = *g ccr(l) exp ( ^ ) - *g-Ccr(2) exp

At rest, no current passes through the membrane and thus the material fluxof chloride ions compensates the material flux of sodium and potassiumions:

• W + . / K + =Ja- (6.5.2)Equations (6.5.1) and (6.5.2) yield the membrane rest potential in the form

. , _RT *£cK+(l) + *£+cNa+(A * M - F l n *£cK+(2) + ^a +cN a +(

462

Ion transport is characterized by conditional rate constants &£+, k^a+ and&a- which can be identified with the permeabilities of the membrane forthese ions.

These relationships can be improved by including the effect of theelectrical double layer on the ion concentration at the membrane surface(cf. page 444 and Section 5.3.2).

Equation (6.5.3) is identical with the relationship derived by Hodgkin,Huxley and Katz. It satisfactorily explains the experimental values of themembrane rest potential assuming that the permeability of the membranefor K+ is greater than for Na+ and Cl~~, so that the deviation of A$M fromthe Nernst potential for K+ is not very large. However, the permeabilitiesfor the other ions are not negligible. In this way the axon at rest would losepotassium ions and gain a corresponding concentration of sodium ions. Thisdoes not occur because of the action of Na+,K+-ATPase, transferringpotassium ions from the intercellular liquid into the axon and sodium ions inthe opposite direction, through hydrolysis of ATP.

When the nerve cells are excited by an electric impulse (either 'natural',from another nerve cell or another site on the axon, or 'artificial', from anelectrode), the membrane potential changes, causing an increase in thefrequency of opening of the gates of the sodium channels. Thus, the flux ofsodium ions increases and the membrane potential is shifted towards theNernst potential value determined by sodium ions, which considerablydiffers from that determined by potassium ions, as the concentration ofsodium ions in the extracellular space is much greater than in theintracellular space, while the concentration ratio of potassium ions is theopposite. The potential shift in this direction leads to a further opening ofthe sodium gates and thus to 'autocatalysis' of the sodium flux, resulting in aspike which is stopped only by inactivation of the gates.

In spite of the fact that the overall currents flowing across the cellmembranes consist of tiny stochastic fluctuating components, the resultingdependences, as shown in Figs 6.19 and 6.20, are smooth curves and can beused for further analysis (the situation is, in fact, analogous to most ofphenomena occurring in nature). Thus, the formation of a spike can beshown to be a result of gradual opening and closing of very many potassiumand sodium channels (Fig. 6.24).

According to Fig. 6.17 the nerve cell is linked to other excitable, bothnerve and muscle, cells by structures called, in the case of other nerve cells,as partners, synapses, and in the case of striated muscle cells, motor end-plates (neuromuscular junctions). The impulse, which is originally electric, istransformed into a chemical stimulus and again into an electrical impulse.The opening and closing of ion-selective channels present in these junctionsdepend on either electric or chemical actions. The substances that are activein the latter case are called neurotransmitters. A very important member ofthis family is acetylcholine which is transferred to the cell that receives thesignal across the postsynaptic membrane or motor endplate through a

463

50

coo

- 5 0

sodium potential

membrane potential

potassium

potential

Fig. 6.24 A hypothetic scheme of the time behaviour of thespike linked to the opening and closing of sodium andpotassium channels. After longer time intervals a temporaryhyperpolarization of the membrane is induced by reversedtransport of potassium ions inside the nerve cell. Nernstpotentials for Na+ and K+ are also indicated in the figure.

(According to A. L. Hodgkin and A. F. Huxley)

specific channel, the nicotinic acetylcholine receptor. This channel has beeninvestigated in detail, because, among other reasons, it can be isolated inconsiderable quantities as the membranes of the cells forming the electricorgan of electric fish are filled with this species.

Acetylcholine, which is set free from vesicles present in the neighbour-hood of the presynaptic membrane, is transferred into the recipient cellthrough this channel (Fig. 6.25). Once transferred it stimulates generationof a spike at the membrane of the recipient cell. The action of acetylcholineis inhibited by the enzyme, acetylcholinesterase, which splits acetylcholine tocholine and acetic acid.

Locomotion in higher organisms and other mechanical actions are madepossible by the striated skeletal muscles. The basic structural unit of musclesis the muscle cell—muscle fibre which is enclosed by a sarcoplasmaticmembrane. This membrane invaginates into the interior of the fibre throughtransversal tubules which are filled with the intercellular liquid. The insideof the fibre consists of the actual sarcoplasm with inserted mitochondria,sarcoplasmatic reticulum and minute fibres called myofibrils, which are theorgans of muscle contraction and relaxation. The membrane of thesarcoplasmatic reticulum contains Ca2+-ATPase, maintaining a concentra-tion of calcium ions in the sarcoplasm of the relaxed muscle below10~7mol • dm~3. Under these conditions, the proteins, actin and myosin,forming the myofibrils lie in relative positions such that the muscle isrelaxed. When a spike is transferred from the nerve fibre to the sarcoplas-matic membrane, another spike is also formed there which continues

464

- 8.5 nm

4 nm

' 8nm

5 nm

Fig. 6.25 The nicotinic acetylcholine receptorin a membrane. The deciphering of the struc-ture is based on X-ray diffraction and electronmicroscopy. (According to Kistler andcoworkers)

through the transverse tubules to the membrane of the sarcoplasmaticreticulum, increasing the permeability of the membrane for calcium ions byfive orders within 1 millisecond, so that the concentration of Ca2+ ions inthe sarcoplasm increases above 10~3 mol • dm~3. This produces a relativeshift of actin and myosin molecules and contraction of the muscle fibre.After disappearance of the spike, Ca2+-ATPase renews the original situa-tion and the muscle is relaxed.

6.5.2 Membrane principles of bioenergetics

The basic process by which living organisms obtain energy isphotosynthesis in green plants, yielding carbonaceous substances (fuels) andoxidative phosphorylation in all organisms, in which the Gibbs energyobtained by oxidation of these carbonaceous substances with oxygen isconverted into the energy of the polyphosphate bond in adenosinetri-phosphate (ATP) through the reaction of adenosine diphosphate with di-hydrogen phosphate ions (PinOrg)-

This process occurs in the mitochondria, organelles present in the cells ofall multicellular organisms (see Figs 6.8 and 6.26). Mitochondria have twomembranes. The invaginations of the internal membrane into the innerspace of the organelle (matrix space) are termed crests (from the Latin,cristae).

The oxidation of the carbonaceous substrate with oxygen cannot occurdirectly as the released energy would be converted into heat, but mustproceed almost reversibly, through a number of poorly water-soluble redox

465

outer membrane

inner membrane

matrix space

intracristal space

8

Fig. 6.26 Section of a mitochondrion

systems forming an electron transfer chain located in the internal membraneof the mitochondrion. They include the system of the oxidized and reducedforms of nicotinamide adenine dinucleotide (NAD+/NADH), several fla-voprotein systems, especially FMN (flavinmononucleotide), Fe-S proteins(where the iron ions are bound in a complex with inorganic sulphide,cystein and a protein), coenzyme Q (ubiquinone) and several cytochromes.These redox systems are bound to phospholipids and other molecules inseveral integrated systems depending on their redox potential (see Table6.2). The redox potentials of these systems, measured in solution, can bequite different from the potentials in the mitochondrial membrane. Thesequence of the redox components in this electron transfer permits theoxidation of the carbonaceous substrates to occur 'vectorially', i.e. thechemical change is oriented in space.

The theory of the accumulation of the Gibbs energy of oxidation ofcarbonaceous substrates in the form of the bond of the phosphate ion to

Table 6.2 Apparent formal redox potentials of systems present in the electron-transfer chain (pH = 7). It should be noted that the potential values were obtained inthe homogeneous phase. Due to stabilization in a membrane, the oxidation-reduction properties vary so that the data listed below are of orientation character

only

Oxidized form

o2Cytochrome a3 (ox)Cytochrome c (ox)Cytochrome b (ox)Coenzyme QFlavinadenin-

dinucleotide (FAD)FumarateNicotinamide adenin-

dinucleotide (NAD+)

Reduced form

H2OCytochrome a3 (red)Cytochrome c (red)Cytochrome b (red)Coenzyme QH2

FADH2Succinate

NADH

£°'(pH = 7)(V)

0.80.50.220.070.10

-0.03-0.03

-0.32

466

adenosine diphosphate (oxidative phosphorylation) must explain the follow-ing facts:

(a) A topologically closed vesicle (a cell or an organelle) which is notpermeable to passively transported H+ is always necessary for transfor-mation of the chemical energy of the substrate to the chemical energy ofATP.

(b) This process is connected with a change in the pH difference betweenthe intracristal space and the matrix space of the mitochondrion as wellas with a change in the membrane potential.

(c) An artificially generated difference in the electrochemical potential ofH3O+ between the intracristal space and the matrix space can result inATP synthesis.

(d) Uncoupling of oxidative phosphorylation (i.e. the process in which thesubstrate is oxidized but no ATP is synthesized) occurs in the presenceof certain proton and alkali metal ion carriers.

P. Mitchell (Nobel Prize for Chemistry, 1978) explained these facts by hischemiosmotic theory.! This theory is based on the ordering of successiveoxidation processes into reaction sequences called loops. Each loop consistsof two basic processes, one of which is oriented in the direction away fromthe matrix surface of the internal membrane into the intracristal space andconnected with the transfer of electrons together with protons. The secondprocess is oriented in the opposite direction and is connected with thetransfer of electrons alone. Figure 6.27 depicts the first Mitchell loop, whosefirst step involves reduction of NAD+ (the oxidized form of nicotinamideadenosine dinucleotide) by the carbonaceous substrate, SH2. In this process,two electrons and two protons are transferred from the matrix space. Theprotons are accumulated in the intracristal space, while electrons aretransferred in the opposite direction by the reduction of the oxidized formof the Fe-S protein. This reduces a further component of the electrontransport chain on the matrix side of the membrane and the process isrepeated. The final process is the reduction of molecular oxygen with thereduced form of cytochrome oxidase. It would appear that this reactionsequence includes not only loops but also a proton pump, i.e. an enzymaticsystem that can employ the energy of the redox step in the electron transferchain for translocation of protons from the matrix space into the intracristalspace.

The energy obtained by oxidation of the substrate with oxygen throughthe electron transport chain is thus accumulated as a difference in theelectrochemical potential for H+ between the intracristal and matrix spaces.

tThe term 'chemiosmotic' theory is not very suitable; 'electrochemical membrane theory'would probably be better, but the former has become accepted. Similarly, Mitchell's term'protonmotive' force is rather unfortunate for the potential difference produced by the transferof a proton across the mitochondrion membrane. Even this term, imitating 'electromotive'force (retained only because of tradition), seems rather awkward.

467

2H*

g

NAD*

2H*

Fig. 6.27 An example of Mitchell's 'loops'.The substrate SH2 is oxidized by NAD+

(nicotinamideadeninedinucleotide) and the re-duced form transports two protons and twoelectrons, of which two protons remain in theintracristal space and two electrons are trans-ported back by the Fe-S protein to reduceFMN (flavinmononucleotide), the reducedform of which transports two protons and two

electrons in the opposite direction

This can be used in several ways. H+-ATPase plays a key role in oxidativephosphorylation (ATP synthesis) as it transfers protons back into the matrixspace with simultaneous synthesis of ATP (with temporary enzyme phos-phorylation, cf. page 451).

In the presence of certain substances, oxidative phosphorylation isuncoupled, i.e. while the carbonaceous substrate is oxidized, ATP is notsynthesized, but can even be hydrolysed to ADP and a phosphate ion. Theuncouplers of oxidative phosphorylation belong to the two groups ofionophores, one of which transfers protons across the mitochondrialmembrane and the other alkali metal ions. The first group includes thesemihydrophobic anions of weak acids such as 2,4-dinitrophenolate, dicu-murate, azide, etc. By resonance effects resulting in spreading of thenegative charge over the whole structure of these anions, they can passthrough the lipid part of the inner membrane of the mitochondrion in thedirection of increasing electric potential into the intracristal space, withhigher concentration of hydrogen ions. They are then converted intothe undissociated form of the corresponding acid, which is transportedin the opposite direction into the matrix space, where protons obtained in

468

the intracristal space are dissociated. The transport of the anions and of theundissociated acid is passive and decreases both the membrane potentialand the activity of hydrogen ions in the intracristal space, so that the energyobtained in the oxidation of the carbonaceous substrate is dissipated asheat. The decrease in the hydrogen ion activity in the intracristal space leadsto the opposite function of H+-ATPase resulting in ATP hydrolysis.

The second group of uncouplers includes macrocyclic complexing agentsof the valinomycin type (see page 445), which have the strongest effect in thepresence of potassium ions. The hydrophobic (lipophilic) ionophore freelydiffuses through the lipid part of the mitochondrial membrane. It bindspotassium ions on the intracristal side, which are then passively transportedinto the matrix space in the complex, simulating proton transport. Theexplanation of the effect of uncouplers strongly supports the Mitchell theoryof oxidative phosphorylation.

The difference in the hydrogen ion electrochemical potential, formed inbacteria similarly as in mitochondria, can be used not only for synthesis ofATP but also for the electrogenic (connected with net charge transfer)symport of sugars and amino acids, for the electroneutral symport of someanions and for the sodium ion/hydrogen ion antiport, which, for example,maintains a low Na+ activity in the cells of the bacterium Escherichia coli.

It should be noted that, while the Mitchell theory explains some of thebasic characteristics of oxidative phosphorylation, a number of phenomenaconnected with this process remain unclear from both the biochemical andthe electrochemical points of view. Biochemical processes in the mitochon-drial membrane are still far from being clarified completely. Although theMitchell theory is called electrochemical, it is obvious that it was notformulated by an electrochemist. It employs only thermodynamic quantitiesand does not consider the kinetics of the individual steps. Charge balance isneglected in charge transfer across the mitochondrial membrane, although itis obvious that part of the transferred charge must be consumed in chargingthe electrical double layer at the membrane surface, which is verydiversified; because of the presence of cristae the capacity of the doublelayer is large. The role of transferred protons in acid-base equilibria is alsonot clear. Nonetheless, the Mitchell theory has been a great step forward.

The second basic bioenergetic process consists of consumption of solarradiation as a source of energy in the organisms. One of the ways ofemploying solar radiation energy in the organism has already beenmentioned in connection with the explanation of the function of the purplemembrane of Halobacterium halobium (page 450); however, a far moreimportant means of employing this energy for bioenergetic purposes isphotosynthesis

6CO 2 + 6H 2 O chlo^ov

phyl, > C 6 H 1 2 O 2 + 6O 2 (6.5.4)

469

while the process in photosynthetic bacteria is

6CO2 + 12H2S bactcrioc1rophyll> C6H12O6 + 12S + 6H2O (6.5.5)

However, process (6.5.5) cannot be a universal photosynthetic processbecause H2S is unstable and is not available in sufficient quantities in nature.Water is the only substance that can be used in the reduction of carbondioxide whose presence in nature is independent of biological processes.

Photosynthesis in green plants occurs in two basic processes. In the dark(the Calvin cycle) carbon dioxide is reduced by a strong reducing agent, thereduced form of nicotinamide adeninedinucleotide phosphate, NADPH2,with the help of energy obtained from the conversion of ATP to ADP:

6CO2 + 12NADPH2 + 6H2O ^ ^ >C6H12O6 + 12NADP

ATP ADP + Pinorg

The process occurring on the light is a typical membrane process. Themembrane of the thylakoid, an organelle that constitutes the structural unitof the larger photosynthetic organelle, the chloroplast, contains chlorophyll(actually an integrated system of several pigments of the chlorophyll andcarotene types) in two photosystems, I and II, and an enzymatic systempermitting accumulation of light energy in the form of NADPH2 and ATP(this case of formation of ATP from ADP and inorganic phosphate istermed photophosphorylation).

Figure 6.28 shows how the process occurring on the light proceeds in thetwo photosystems. Absorption of light energy involves charge separation inphotosystem II, i.e. formation of an electron-hole pair. The hole passes tothe enzyme containing manganese in the prosthetic group, which thusattains such a high redox potential that it can oxidize water to oxygen andhydrogen ions, set free on the internal side of the thylakoid membrane. Theelectron released from excited photosystem II reacts with plastoquinone(plastoquinones are derivatives of benzoquinone with a hydrophobic alkylside chain), requiring transfer of hydrogen ions from the external sideof the thylakoid membrane. The plastohydroquinones produced reducethe enzyme, plastocyanine, releasing hydrogen ions that are trans-ferred across the internal side of the thylakoid membrane. On excita-tion by light, chlorophyll-containing photosystem I accepts an electronfrom plastocyanine by mediation of further enzymes. This high-energyelectron then reduces the Fe-S protein, ferredoxin, which subsequentlyreduces NADP via further enzymes. Part of the energy is thus accumulatedin the form of increased electrochemical potential of the hydrogen ionswhose concentration—as has been seen—increases inside the thylakoid. Thereverse transport of hydrogen ions provides H+-ATPase with energy forsynthesis of ATP. In contrast to cellular respiration, where the energy

777H Ferredoxin

Chl-aT

llMimhiiliL

iuiiiiiiii ill iiiiiiiWHiiiiimiill

H+

H+ ATPase

Plastocyanin

e) S

Mn-proteincomplex

///////////////I

71X,,

777m Him

hll/llll/MN tilllll

SYSTEM I H +

Fig. 6.28 The light process of photosynthesis. (According to H. T. Witt)

H+

SYSTEM E

471

obtained in the oxidation of the carbonaceous substrate accumulates in theform of a single 'macroenergetic' compound of ATP, two energy-richcompounds are formed during photosynthesis, NADPH2 and ATP, both ofwhich are consumed in the synthesis of sugars. The subsequent steps appearto be purely biochemical in character and therefore will not be consideredhere.

In a partly biological, partly artificial model (page 397) reducedanthraquinone-2-sulphonate plays the role of NAD+ and tetramethyl-/?-phenylenediamine that of plastoquinones.

References

Baker, P. F. (Ed.), The Squid Axon. Current Topics in Membranes and Transport,Vol. 22, Academic Press, Orlando, 1984.

Barber, J. (Ed.), Topics in Photosynthesis, Vols. 1-3, Elsevier, Amsterdam, 1978.Briggs, W. R. (Ed.), Photosynthesis, A. R. Liss, New York, 1990.Calahan, M., Molecular properties of sodium channels in excitable membranes, in

The Cell Surface and Neuronal Function (Eds C. W. Cotman, G. Poste and G. L.Nicholson), P. I, Elsevier, Amsterdam, 1980.

Catteral, W. A., The molecular basis of neuronal excitability, Science, 223, 653(1984).

Cole, K. S., Membranes, Ions and Impulses, University of California Press,Berkeley, 1968.

Conti, F., and E. Neher, Single channel recordings of K+ currents in squid axons,Nature, 285, 140 (1980).

Cramer, W. A., and D. B. Knaff, Energy Transduction in Biological Membranes, ATextbook of Bioenergetics, Springer-Verlag, Berlin, 1989.

French, R. J., and R. Horn, Sodium channel gating: models, mimics and modifiers,Ann. Rev. Biophys. Bioeng., 12, 319 (1983).

Govindjee (Ed.), Photosynthesis, Vols. 1 and 2, Academic Press, New York, 1982.Hagiwara, S., and L. Byerly, Calcium channel, Ann. Rev. Neurosci., 4, 69 (1981).Hille, B., Ionic Channels of Excitable Membranes, Sinauer, Sunderland, 1984.Katz, B., Nerve, Muscle and Synapse, McGraw-Hill, New York, 1966.Keynes, R. D., The generation of electricity by fishes, Endeavour, 15, 215 (1956).Keynes, R. D., Excitable membranes, Nature, 239, 29 (1972).Lee, C. P., G. Schatz, and L. Ernster (Eds), Membrane Bioenergetics, Addison-

Wesley, Reading, Mass., 1979.Mitchell, P., Davy's electrochemistry—Nature's protochemistry, Chemistry in

Britain, 17, 14 (1981).Mitchell, P., Keilin's respiratory chain concept and its chemiosmotic consequences,

Science, 206, 1148 (1979).Moore, J. W. (Ed.), Membranes, Ions and Impulses, Plenum Press, 1976.Nicholls, D. G., Bioenergetics, An Introduction to the Chemiosmotic Theory,

Academic Press, Orlando, 1982.Sakmann, B., and E. Neher (Eds), Single-Channel Recordings, Plenum Press, New

York, 1983.Sauer, K., Photosynthetic membranes, Ace. Chem. Res., 11, 257 (1978).Skulatchev, V. P., Membrane bioenergetics—Should we build the bridge across the

river or alongside of it?, Trends Biochem. Sci., 9, 182 (1984).Stein, W. D. (Ed.), Current Topics in Membranes and Transport, Vol. 21, Ion

Channels: Molecular and Physiological Aspects, Academic Press, Orlando, 1984.

472

Stevens, C. F., Biophysical studies in channels, Science, 225, 1346 (1984).Stroud, R. M., and J. Tinner-Moore, Acetylcholine receptor, structure, formation

and evolution, Ann. Rev. Cell BioL, 1, 317 (1985).Tsien, R. W., Calcium channels in excitable cell membranes, Ann. Rev. Physiol.,

45, 341 (1983).Ulbricht, W., Ionic channels and gating currents in excitable membranes, Ann. Rev.

Biophys. Bioeng., 6, 7 (1977).Urry, D. W., A molecular theory of ion-conducting channels: A field dependent

transition between conducting and non-conducting conformations, Proc. Natl.Acad. Sci. USA, 69, 1610 (1972).

Van Dam, K., and B. F. Van Gelder (Eds.), Structure and Function of EnergyTransducing Membranes, Elsevier, Amsterdam, 1977.

Witt, H. T., Charge separation in photosynthesis, Light-Induced Charge Separationin Biology and Chemistry (ed. H. Gerischer and J. J. Katz), Verlag Chemie,Weinheim, 1979.

Appendix A

Recalculation Formulae forConcentrations and Activity

Coefficients

(a) Concentrationterm

Mole fraction

In dilutesolution

Molality

In dilutesolution

Molar concentration

In dilutesolution

(b) Activitycoefficient

Y±,x

Y±,m

Y±,c

Mo + vx i Mo

Mo

Afo + M + vAfo)*,

~M^

Y±.x

y±tX(\ - vxY)

p0/ MA

Expressed in

m1M0

1 + vm1M0

/n,Af0

m,

pwi\1 + YYl j Mj

mxp{)

y ±,m

c M

P + ( 7 ^ M l ) C l

Po

ci

Po

±>C Pop-c,Af,

±>C Po

y±,r

Mo and M! denote molar mass (kg • mol ') of the solvent and of the solute, respectively, v isthe number of ions formed by an electrolyte molecule on dissolution, p and p0 denote thedensity of the solution and of the pure solvent, respectively, and nx and n0 are the amount ofthe solute and of the solvent (mol), respectively. The above relationships also hold for the ionicquantities m+, m_> c+, etc., where m,, c, and xx are replaced by m+ = v+m^, m_ = v_mx,c+ = v+clt etc.

473

Appendix B

List of Symbols

A affinity, surfacea activity, effective ion radius, interaction coefficientc concentrationC differential capacity, dimensionless concentrationD diffusion coefficient, relative permittivityE electromotive force (EMF), electrode potential, electric field strength,

energy; Em half-wave potential, Subscript standard electrodepotential, Epzc potential of zero charge, Ep potential of idealpolarized electrode, potential energy

e proton chargeF Faraday constantG Gibbs energy, conductanceg gaseous phaseg gravitation constantH enthalpyH° acidity function/ electric current, ionization potential/ , J flux of thermodynamic quantity/', j current densityK equilibrium constant, integral capacityk Boltzmann constant, rate constant, k.a anodic electrode reaction rate

constant, adsorption rate constant, kc cathodic electrode reactionrate constant, kd desorption rate constant, k^ conditional electrodereaction rate constant

L phenomenological coefficientLD Debye length/ length1 liquid phaseM relative molar mass ('molecular weight'), amount of thermodynamic

quantitym molality, flowrate of mercurym metallic phase, membraneAf number of particles; NA Avogadro constantn amount of substance (mole number), number of electrons ex-

changed, charge number of cell reaction

474

475

P solubility productR electric resistance, gas constantr radiuss overall concentration, number of componentss solid phase5 entropyT absolute temperaturet time, transport (transference) number; tx drop timeU electrolytic mobility, internal energy, potential difference (Voltage')u mobilityV volume, electric potentialv molar volume, velocityX, X thermodynamic forcex molar fraction, coordinateZ impedancez charge number

a dissociation degree, charge transfer coefficient, real potential/? buffering capacity, adsorption coefficienty activity coefficient, interfacial tensionF surface concentration, surface excessAG reaction Gibbs energy changeAH activation enthalpy (energy) of electrode reactionA<t> Galvani potential differenceAt/; Volta potential difference6 thickness of a layer, deviationE permittivity, electron energy, eF Fermi level energy£ electrokinetic potentialrj overpotential, viscosity coefficient0 relative coverage6 wetting angleK conductivity, mass transfer coefficientA molar conductivityA ionic conductivity, fugacity[i chemical potential, reaction layer thickness; fi electrochemical

potentialv kinematic viscosity, number of ions in a molecule, radiation

frequencyn osmotic pressurep density, space charge densityo surface charge density, rate of entropy productionT transition time, membrane transport number0 inner electrical potential, osmotic coefficientX surface electrical potential

476

ip electrical potential, outer electrical potential, density of a thermo-dynamic quantity

co frequency, angular frequencyO dissipation function; Oe electron work function

Subscripts and superscripts of these symbols give reference to chemicalspecies, charge, standard state, etc., of quantities concerned. Symbols thatseldom occur have not been included in the above list, their meaning havingbeen given in the text.

Index

A.c. polarography, 303, 296Acetylcholine, determination, 428

neurotransmitter, 462-461Acid-base theory,

Arrhenius, 9-13, 45, 52Br0nsted, 45-55generalized, 59-61HSAB, 61Lewis, 59-60

Acidity function Ho, 65-66Acidity scale, 63Acrylonitrile, electrochemical reduction,

387Activation enthalpy, of electrode

reaction, 255, 257, 265-266, 271-272, 389

Active transport, 449-452Activity, definition, 5-6

mean, 8molal, 7rational, 7

Activity coefficient, 6-8, 29-45definition, 6-8in electrolyte mixture, 41-43methods of measurement, 44-45molal, 7of electrolyte, 29-45potentiometric measurement, 195rational, 7recalculation formulae, 473

Ad-atom, 306, 371Adiabatic approximation, 267Adiabatic electron transfer, 273Adiponitrile, electrochemical synthesis,

387Adsorption,

electrostatic, 199Gibbs, 205in electrode process, 250, 352-368of surfactants, 224-231, 364-367specific, 219-224

Adsorption isotherm,Frumkin, 227Gibbs, 205Langmuir, 226-227linear, 226Temkin, 228

Alkali metals, in liquid ammonia, 20-21Amalgam electrode, 171-172Amino acid, dissociation, 70-73Ammonolysis, 54Amphion, 71Amphiprotic solvent, 46

protolysis, 53, 58Ampholyte, 70-74Anion, 1, 246Anode, 1, 146Anodic current, definition, 246Antiport, 450Aromatic hydrocarbons, electrochemical

reduction, 385Aromatic nitrocompounds, reduction,

see NitrocompoundsArrhenius electrolyte theory, 9-13, 45,

52Ascorbic acid oxidation, 350Association constant, 20-27Atomic force microscope, 341ATPase, H+, 451, 467

Na+,K+, 451-453, 454, 462Ca2+, 463, 464

Attenuated total reflection, 332Auger electron spectroscopy, 338Autoionization, see Self-ionizationAuxilliary electrode, 291Axon, 454

Bacteriorhodopsin, 451Bacterium electrode, 432Band gap, 89Band theory, 87-89Barrier film, see Continuous film

477

478

Base electrolyte, 116Bates-Guggenheim equation, 37Becquerel effect, 391Beer-de Nora anodes, 312Bilayer lipid membrane, 429-442, 448Bioelectrochemistry, 410Biological membrane, 453-471

composition, 434-437structure, 438-439

Biosensor, 431-432Bipolaron, 324, 325Bjerrum theory, of ion association, 24-

26Blander equation, 27BLM, see Bilayer lipid membraneBoltzmann distribution, 24Born equation, of solvation, 16-17

refinement, 17Brewster angle, 331Br0nsted acid-base theory, see Acid-

base theory, Br0nstedBuffer, 55-57

buffering capacity, 56

Cadmium cyanide, electrochemicalreduction, 250

Cadmium sulfide electrode, 309Calomel electrode, 176-177Calorimetric method, determination of

work function, 157Capillary electrometer, 232-233Carbon, 313-316

electrochemical carbon, 315glass-like carbon, 314-315glassy carbon, 314-315surface oxides, 314vitreous carbon, 314see also Graphite

Catalytic current, 350Cathode, 1,246Cathodic current, definition, 246Cation, 1Cell reaction, definition, 160Cell wall, 438Channel, 438, 445, 447-448, 458-464Charge carrier, 87Charge number, definition, 3Charge transfer coefficient, 256, 273Charging current, 234Chemical flux, 81,87Chemical potential, definition, 4-6, 146Chemical reactions in electrode process,

250-251, 344-352Chemically modified electrodes, 319-323

Chemiosmotic theory, 466Chemisorption, 224, 278, 352Chiral electrode, 319Chlorine electrode, 174-175Chlorine production, 312Chronoamperometry, 283Chronopotentiometry, 300Clark oxygen sensor, 432Coion, 416, 419Compact layer, 278Complex formation, 55

kinetics, 349Concentration cell, 167, 171-172Concentration polarization, 252, 289-

290Conditional potential, see Formal

potentialConductimetry, see ConductometryConduction band, 87-88Conduction of electric current, 81, 84-

86in electrolytes, 85-86, 90-104

Conductivity, 2, 85, 90, 100equivalent, 91ionic, 90, 102limiting, molar, 2molar, 2of electrolytes, 79, 90-101

Conductivity, membrane, 440Conductometry, 100-101Conductors, classification, 87-89Conjugate pair, 46Conjugation entropy model, 129Contact angle, see Wetting angleContact potential, 153, 155Continuity equation, 83Continuous film, 378Controlled potential methods, 293Convection, 81-87

definition, 81forced, 135-137natural, 81, 137

Convective diffusion, 135-147to growing sphere, 139-141, 281to rotating disc, 138-139, 143, 284-

285Conventional rate constant, see

Electrode reaction rate constant,conventional

Convolution analysis, 288-289Corrosion, 381-383

local cells, 383Cottrell equation, 281Coulometry, 303-305

479

Coulostatic method, 300-301Counterion, 416, 419Coverage (relative), 226-228, 230, 361-

362, 366-367determination, 235, 366

Crystallite, 376Curie principle, 81, 450Cyclic voltammetry, 294-296, 320, 356

Daniell cell, 159-160Debye-Falkenhagen effect, 100Debye-Huckel limiting law, 9, 29-34Debye-Hiickel theory, 29-39

more rigorous treatment, 34-37Debye length, 32, 36, 216

in semiconductors, 236-237Dendrites, 377, 379Depletion layer, 239

see also Space charge regionDepolarization, 455Desorption potential, 224Dialysis, 422Differential capacity, 207-208, 213, 216-

219, 224, 225, 229, 231, 234-235determination, 234of semiconductor electrode, 240

Differential pulse polarography, 299Diffuse electric layer, 200-201, 211,

213-217, 272, 274-278at ITIES, 240-241at membrane, 443-444in semiconductor, see Space charge

regionDiffusion, definition, 80-81

in electrolyte solution, 104-105, 110-117

in flowing liquid, 134-147in solids, 124-126linear, 105-109spherical, 109-110steady state, 109, 284-287surface, 372transient, 104-110

Diffusion coefficient, determination,117-120

in electrolyte solution, 115-117Diffusion coefficient theory, 121-126Diffusion, lateral, 438Diffusion law, see Ficks's lawDiffusion layer, 107Diffusion potential, 111-115Dimensionally stable anodes, 312Dipole-dipole interaction, 18

in water, 18

Discontinuous film, 377Disproportionation, 181-182, 350Dissipation function, 84Dissociation constant, 10-12, 50-52

in non-aqueous solvents, 53potentiometric determination, 195-

197Dissociation, electrolytic, 1Distance of closest approach, 25, 34Distribution potential, 190DME, see Dropping mercury electrodeDonnan equilibrium, 414Donnan potential, 412-414, 417-418Double layer, see Electrical double layerDriving force, 80Dropping mercury electrode, 233-234,

295-299DSA, see Dimensionally stable anodesDupre equation, 204

Einstein-Smoluchowski equation, 121Electrical double layer, 145, 198-243,

274at ITIES, 241-242at metal-electrolyte solution

interface, 198-235at semiconductor-electrolyte solution

interface, 235-240Electroactive substance, 246-247Electrocapillarity, 198, 203-213Electrocapillary curve, 207, 208, 225Electrocatalysis, 352-361, 364Electrochemical potential, definition,

146Electrochemical power source, 168Electrocrystallization, 368, 372Electrode, first kind, 169-175

second kind, 169, 175-177Electrode material, 291, 305-328Electrode potential, absolute, 168-169

additivity, 150-181formal, 167, 178in fused salts, 175in organic solvents, 184-188standard, 163-164, 171-172

Electrode preparation, 307-308Electrode process, 246

mechanism, 250Electrode reaction

at semiconductor electrode, 248-249definition, 246effect of electrical double layer, 272,

274-278irreversible, 257, 283, 286

480

Electrode reaction (cont.)of complexes, 347, 349two-step, 262-264

Electrode reaction order, 254Electrode reaction rate, 233-264

effect of diffusion, 279-290elementary step, theory, 266-279phenomenological theory, 254-266

Electrode reaction rate constantconditional, 257-259conventional, 262

Electrodialysis, 424-425Electrokinetic phenomena, 241-243Electrokinetic potential, 242Electrolyte, 1-2

strong, 2weak, 2

Electrolytic cell, 245Electrolytic mobility, definition, 86Electromotive force, definition, 159

measurement, 191-192standard, 163temperature quotient, 162thermodynamic theory, 160-167

Electron acceptor, 89Electron donor, 89Electron hopping, 323Electron microprobe, 338Electron paramagnetic resonance, see

Electron spin resonanceElectron spectroscopy, 337Electron spin resonance, 330Electron transfer, 266, 269Electron transfer reaction, see Electrode

reactionElectron transfer theory, 266-274Electron work function, see Work

functionElectroneutrality condition, 3Electroosmotic flow, 242, 419-420Electroosmotic pressure, 423Electrophilic substance, 60Electrophoresis, 243-244Electrophoretic effect, 93-95Electroreflectance, 333Ellipsometry, 335Emeraldine, 327EMF, see Electromotive forceEnergy profile, 267, 270-271Entropy production, 84Enzyme electrode, 432Epitaxy, 376ESC A, see Electron spectroscopy

ESTM, see Scanning tunnellingmicroscopy

Ethanol, as solvent, 53EXAFS, see Extended X-ray absorption

fine structureExchange current, 257-259, 262Excited state, 393

acid-base properties, 58-59Exponential law, 381Extended X-ray absorption fine

structure, 337External reflectance, 332

Facetting, 377Faradaic current, 249Faradaic impedance, 301-304Faraday law, 85, 90, 249-250, 279Fermi energy, 148-151, 247, 398Fermi function, 148Fick's law, first, 86Fick's laws, 105-106Field dissociation effect, 98-100Film growth, 377-378Flade potential, 378-380Flatband potential, 237, 238Fluid mosaic, 36Fluorescein, 68Fluorographite, 318Fluorolysis, 54Fluoropolymers, electrochemical

carbonization, 316Flux, of thermodynamic quantity, 80-83Forbidden band, 88-89

see also Band gapFormal potential, 167, 178

apparent, 178Formaldehyde, electrochemical

reduction, 349Formic acid, oxidation, 387-388Fourier's equation, 86Fourier transformation, 333Fourier transformation infrared

spectroscopy, see SNIFTIRSFranck-Condon principle, 268Free convection, see Convection, naturalFree volume model, 129Frenkel defect, 124-126Frenkel mechanism, 125Fresnel equation, 331Frumkin theory, 278Fugacity, 6Fused salts, 13-14, 127, 175

protolysis, 58-59

481

Gallium electrode, 305Galvanic cell, 157-169

with transport, 167without transport, 167

Galvani potential difference, 147-153,164,167

Galvanostatic method, 300Gas probe, 431Gate, 458-462Gauss-Ostrogradsky theorem, 83Geometric surface, 246Germanium, 89

electrode, 308, 332Gibbs adsorption equation, 205Gibbs energy, partial molar, 4Gibbs- Helmholtz equation, 162Gibbs-Lippmann equation, 207, 226Gibbs transfer energy, 62-63, 185-190Gierke model, of Nafion structure, 132-

133Glass electrode, 428-431Glass-like carbon, see CarbonGlassy carbon, see CarbonGlycocalyx, 439Gold electrode, 182, 308Gouy-Chapman theory, 214-217Gouy layer, see Diffuse electric layer,Gramicidin A, 447-449, 458Graphene,316Graphite, 313

exfoliated, 318highly oriented pyrolytic (HOPG),

313Grotthus mechanism, 123Gurevich equation (photoemission), 392

Half-cell reaction, 161Half-crystal position, 305-306Half-wave potential, 282, 286, 287Halobacterium, 451, 468Hard acids and bases, 61Hard sphere approximation, 214Harned rule, 43, 45Heat conduction, 86

definition, 81Helmholtz layer (plane), see Compact

layerHelmholtz-Smoluchowski equation, 420Henderson formula, 113, 418-419Henderson-Hasselbalch equation, 56Henry's law, 5, 186Heyrovsky reaction, 353Hittorf s method, 102

Hittorf s number, 103Hole, 88-89, 273, 247-248HOPG, see GraphiteHSAB, see Acid-base theory, HSABHump, 219Hydration, 15, 19

number, 22-23, 39-41sphere, 19

Hydrodynamic layer, see Prandtl layerHydrogen electrode, 161-165

standard, 166, 173-175Hydrogen, electrode reactions, 353-358,

362-363isotope effect, 358

Hydrogen bonds, 14Hydrogen ion, see Oxonium ionHydrogen peroxide, 359Hydrolysis, of salts, 54-55Hydrophobic interaction, 20Hydroxonium ion, see Oxonium ion,

Ideal polarized electrode, 201, 202, 206,209, 224

IETS, see Inelastic electron tunnellingspectroscopy

Ilkovic differential equation, 140Ilkovic equation, 281, 296-297Impedance electrochemical, 301, 304

Warburg, 303Impedance spectrum, 301-302Indicator, acid-base, 64-69

fluorescence, 68-69determination of cell pH, 68-69

Indicator electrode, see Workingelectrode

Indifferent electrolyte, 43, 116-17, 247,385

Inelastic electron tunnellingspectroscopy, 339

Infrared spectroscopy, 333, 334Inhibition of electrode reactions, 361-

367Inner electrical potential, definition, 146Inner Helmholtz plane, 200, 219, 221,

222Insertion compounds, 316Insertion reactions, see Insertion

compoundsIntegral capacity, 208-209Intercalation, 316

stages, 317Intercalation compounds, see

Intercalation

482

Interface, definition, 144metal-fused salt electrolyte, 242solid electrolyte-electrolyte solution,

241-242Interface of two immiscible electrolyte

solutions (ITIES), 188-190, 200,240-241,249,278,295

charge transfer, 249Interfacial tension, 198, 203-209, 211,

233, 234Interionic interaction, 28-45Internal reflectance 332Interphase, definition, 144-145Interstitial position, 124-125Ion, 1

association, 23-27hydration, 15solvation, 15, 20

Ion-dipole interaction, 18Ion-exchange membrane, 415-425

potential, 417-419transport, 419-425

Ion pair, 23-27Ion product, 47-49

overall, 48Ion scattering spectroscopy, 338Ion-selective electrode, 424-431

calibration, 431Ionic atmosphere, 30-36, 95-100Ionic conductivity, 90, 93, 97Ionic solvent, see Fused saltIonic strength, definition, 9, 32Ionization potential, 151Ionomer, 131Ionophore, 445-447Iron electrode, 379Irreversible thermodynamics, 80-81

membrane process, 421-425, 450-451Isoelectric point, 72-73ISS, see Ion scattering spectroscopyITIES, see Interface of two immiscible

electrolyte solutions

Jellium, 219Junction potential, see Liquid junction

potential

Kink, 305, 306Kohlrausch's law, 79, 92Kolbe reaction, 387-389

Lanthanum trifluoride, 127, 431Laser force microscope, 341Lead dioxide, 311,379

LEED, see Low energy electrondiffraction

Lewis acid-base theory, see Acid-basetheory

Lewis-Sargent equation, 113Limiting current, 280-281, 284-285, 297

kinetic, 345, 349Liquid ammonia, as solvent, 53Liquid junction potential, 111-114

elimination, 114Lithium electrode, 334Local cells, see CorrosionLow energy electron diffraction, 339Luggin capillary, 292-293, 295Luminisence, electrochemically

generated, 331Luther equation, 181Lyate ion, definition, 47Lyonium ion, definition, 47

Macroions, 73Magnetic force microscope, 341Marcus theory, 269-274Mass spectrometry, electrochemical, 340Mass transfer coefficient, 136, 284Mediator, 184, 320Mechanism of electrode process, 250Membrane, conductivity, 449

definition, 411Membrane model, 439-442Membrane pore, 445-449Membrane potential, 411-414, 417-418

axon membrane, 455-463Medium effect, 62Melts, as ion conductors, 127Mercuric oxide electrode, 176Mercury electrode, 211, 218, 295-299,

305, 359-360see also Dropping mercury electrode

Metal, as conductor, 87-88anodic oxidation, 377deposition, 368-383

Metal oxides, 309-313Methanol,

as solvent, 53electrochemical oxidation, 364-365

Microelectrode,ion selective, 426voltammetric, see Ultramicroelectrode

Micropipette, 177Micropotential, 220-221Migration flux, 85, 110Mitchell loop, 466Mitchell theory, 465-468

483

Mitochondrion, 437, 464-468Mixed potential, 381-384, 461Mobility, definition, 86

ionic liquid, 127theory, 120-122

Molality, mean, 4Molar conductivity, 90-93, 98Mossbauer spectroscopy, 337Mott-Schottky equation, 239-240Muscle cell, 463Muscle contraction, 463-464

Nation, 132Nasicon, 125Natural convection, see Convection,

naturalNavier-Stokes equation, 137, 141Nernst-Einstein equation, 86Nernst equation, 166, 171, 280Nernst-Hartley equation, 115Nernst layer, 136, 139, 143Nernst mass, 126Nernst-Planck equation, 111Nernst potential, 412, 462

at ITIES, 188Nerve cell, 454-462Neurotransmitter, 462Nikolsky equation, 428Nitrocompounds aromatic, reduction,

386Nobel Prize, for electrochemistry, 267,

295for membrane biology, 465for physiology and medicine, 461

Noise, 374, 458Non-adiabatic electron transfer, 269Non-faradaic current, 249Non-polarizable electrode, 201, 202Nucleation, 369-375, 378Nucleus, electrocrystallization, 369,

371-375Nyquist diagram, 303

Ohmic potential, 111, 252, 291-292Ohm's law, 2, 84-85, 381Onsager limiting law, 98Optical methods, 328-342Optically transparent electrodes, 312,

330Optoacoustic spectroscopy, see

Photoacoustic spectroscopyOrganic compounds, oxidation, 364,

386-390Organic electrochemistry, 384

Osmosis, 424-425Osmotic coefficient, 8-9

Debye-Huckel theory, 38Osmotic pressure, 8-9, 422, 423Ostwald dilution law, 93, 98OTE, see Optically transparent

electrodesOuabain, 450Outer electrical potential, definition, 146Outer Helmholtz plane, 199, 211, 273,

278Overpotential, 252, 289, 290

concentration, 289transport, 289

Oxidation-reduction electrode, seeRedox electrode

Oxidative phosphorylation, 464-467uncoupling, 466-467

Oxonium ion, definition, 97-98transport, 123-124

Oxygen, electrode reactions, 275-278,358-361

Parabolic law, 380Passivation films, 377Passive transport, 442-449Passivity, 377-378Patch-clamp, 458-459Peak voltammogram, 288-290Peclet number, 142Percolation theory, 130, 134Periodic methods, 301Permittivity, 17Permselective membrane, 416Permselectivity, 132Pernigraniline, 327pH, definition, 50, 63

measurement, 192-194by hydrogen electrode, 173-174

mixed solvents, 53-54non-aqueous medium, 188operational definition, 50, 193practical scale, 193-194

pH range, in water, 53in alcohols, 53

Phenolphthalein, 67Phenomenological coefficient, 80, 422-

425, 450-451Phosphazene, 131Photoacoustic spectroscopy, 335Photocorrosion, 309Photocurrent, 401

quantum yield, 405

484

Photoelectric method of work functiondetermination, 157

Photoelectrochemistry, 390-409Photoelectrolysis, 402, 403Photoemission, electrochemical, 392Photoexcitation, 393-394Photogalvanic cell, 395-397Photogalvanic effect, 390-397Photogalvanovoltaic effect, 391Photophosphorylation, 496Photopotential, 400Photoredox reactions, 394-397Photosynthesis, 469-471Photothermal spectroscopy, 335Photovoltaic cell, 402Photovoltaic effect, 390, 397Pinacol, 250Planck's theory of liquid junction

potential, 114Platinum electrode, 165-166, 177-178,

307-308, 355-358, 361-365, 387-388

hydrogen adsorption, 355-356, 362-363

platinized, 308single crystal, 308, 357

Poggendorf method, 191-192Poisson-Boltzmann equation, 215, 217Poisson distribution, 373Poisson equation, 30, 214, 236, 419Poisson probability relation, 373Polarizable interface, 145Polarization curve, 259, 284

irreversible, 286-287quasirreversible, 287-288reversible, 285

Polarization, of electrode, 252Polarization resistance, 259, 302Polarized radiation, 331, 335Polarography, 295-298Polaron, 324Polyacetylene, 323, 324Polyaniline, 326-328Polyelectrolyte, 73-78

conformation, 74-76dissociation, 77-IS

Polyethylene oxide), 128-129Polyions, 73Polymers

conjugation defects, 324doping (p, n), 323electronically conducting, 323glass transition temperature, 129ion conducting, 128-134

ion coordinating, 128ion exchange, 131ion solvating, 128

Polymer electrolyte, see Polymers, ionconducting

Polymer modified electrodes, 321Polyparaphenylene, 326Poly(propylene oxide), 128Polypyrrole, 326-327Polythiophene, 326-327Potential, chemical, 4, 28Potential of zero charge, see Zero-

charge potentialPotential probe, 177Potential sweep voltammetry, 288-289,

294, 296Potential-decay method, 300Potentiodynamic methods, 295Potentiometry, 181-197Potentiostat, 293-296, 304Potentiostatic methods, 293-296Power spectrum, 374, 448, 458Prandtl layer, 135-136, 141Propylene carbonate, 334Protic solvent, 47Protogenic solvent, 45-46Protolysis, 45, 57-58, 62Protolytic reaction, see ProtolysisProton transport, in water and ice, 123-

124Protophilic solvent, 46

Quinhydrone electrode, 182-184Quinone, electrochemical reduction, 251

Radioactive tracer technique, 328, 342Raman spectroscopy, 335, 336

surface enhanced, 336resonance, 336

Randles-Sevcik equation, 289Random processes, see Stochastic

processesRaoult's law, 5Rate constants, 255, 257, 262, 272, 273-

274Rational potential, 217RBS, see Rutherford backscattering

spectroscopyReaction coordinate, 267, 270-271Reaction layer, 348Real potential, 153Real surface, 246Reciprocity relationship, 80Redox electrode, 177-180, 182-184

485

Reference electrode, 166, 252, 292-293Reflection spectroscopy, 331Refractive index, 332Relaxation effect, 96Reorganization energy, 269-272Resistance, 2Resistivity, 2Resting potential, theory, 461-462Reynolds number, 142-143Ring-disc electrode, see Rotating ring-

disc electrodeRobinson-Stokes equation, 39-41Rotating disc, 138-139, 143Rotating disc electrode, 285, 298-299Rotating ring-disc electrode, 299, 300Roughness factor, 246, 308Ruthenium(II) bipyridine, 394, 396, 405,

406Ruthenium dioxide, 312Rutherford backscattering spectroscopy,

339

Salt bridge, 44, 211Salting-out effect, 22Scanning ion conductance spectroscopy,

342Scanning tunnelling microscopy, 340Scanning tunnelling spectroscopy, 341Schmidt number, 142Schottky barrier, 239Schottky defect, 124-126Schottky mechanism, 126Screw dislocation, 306, 307, 375Secondary ion mass spectroscopy, 338Sedimentation potential, 243Selectivity coefficient, 428Self-ionization, 46-48, 53, 57, 59Semiconductor, 88-89, 248, 308

band bending, 399band theory, 398degenerate, 310electrode, 308, 397-406intrinsic, 88n-type, 88-89, 148photoelectrochemistry, 397-406p-type, 89, 148sensitization, 403

Semipermeable membrane, 410-411Semiquinone, 181Sensitizer, 403SERS, see Raman spectroscopy, surface

enhancedSherwood number, 143Silicon electrode, 308, 309

Silver chloride electrode, 161-166, 175,177

Silver electrode, 306, 358-360, 373Silver iodide, as solid electrolyte, 126-

127Sims, see Secondary ion mass

spectroscopySinglet states, 393SNIFTIRS, 333, 334Sodium channel, 458-462Soft acids and bases, 61Solar energy, photoelectrochemical

conversion, 406-408Solid electrolyte, 124-127Solid polymer electrolyte (SPE) cells,

132Soliton, 324Solubility product, 69-70, 186Solvated electron, 21, 247, 385Solvation, 15-23

in ionic solvents, 21method of investigation, 22-23of ions, 15-23

Solvation energy, 17-18Solvation number, 17, 22-23Solvent,

acid-base properties, 46-47ionic, 13-14molecular, 13structure, 14-15

Solvolysis, 50-51Space charge region, 236-237Sparingly soluble electrolytes, 69-70Specular reflectance, 332Spike, 456-457, 462-463Spiral growth, 375-376Stability constant, 55, 349Steps, 305, 306Stern-Volmer constant, 395STM, see Scanning tunnelling

microscopyStochastic effects (processes), 369, 373-

374,448,461,462Stoichiometric number, of electrode

reaction, 254Stokes law, 95, 121Stokes-Einstein equation, 122Streaming potential, 242Striated skeletal muscle, 463-464Strong electrolyte, see Electrolyte,

strongStructure breaking, 19Sub-lattice, 21, 127

motions, in solid electrolytes, 127

486

Subtractively normalized interfacialFourier transformation infraredspectroscopy, see SNIFTIRS

Sulphide ion, adsorption, 224Surface charge, 199-201, 206-207, 209,

225, 230determination, 234of semiconductor, 237, 239

Surface concentration, 205-206Surface electrical potential, definition,

146-147Surface enhanced Raman spectroscopy,

see Raman spectroscopySurface excess, 206-207, 209, 211, 224-

226Surface modification, 319-323Surface pressure, 226Surface reactions, 350-352Surfactant, 199, 224

as inhibitor, 366-367Symport, 451,467Synapse, 455, 462System trajectory, 267

Tafel equation, 260-262, 275-277, 354,355

Tafel plot, 260Tafel reaction, 353TATB assumption, 187-188, 241Test electrode, see Working electrodeTetrodoxin, 458-459Thermogalvaniccell, 158Thermoionic method, of work function

determination, 157Thionine, 394-396Time-of-relaxation effect, 96-97, 100Tissue electrode, 432Titanium dioxide, 311, 405

photoelectrolysis of water, 402, 403sensitization, 405

Titanium disulphide, 318Transfer activity coefficient, 62Transfer energy, see Gibbs transfer

energyTransference number, 91, 101

Transition time, 109, 283Transmission coefficient, 354Transport number, 101-104Triangular pulse voltammetry, 294, 296Triplet states, 393

Ultramicroelectrode, 110, 292, 298-299Underpotential deposition, 371Uniport, 451Uranyl ion reduction, 350

Vacancies, in crystals, 124Valence band, 88-89Valinomycin, 428, 445-446, 467Valve metals, 311Van't Hoffs factor, 10Vibrating condenser method, 155-156Vitreous carbon, see CarbonVolmer reaction, 353Volmer-Weber equation, 371Volta potential, 147, 153-157Voltage clamp, 294, 457Voltammogram, 284, 285, 288

Wagner parabolic law, 380Walden rule, 123Warburg impedance, see ImpedanceWater structure, 14-16Weak electrolyte, see Electrolyte, weakWeston cell, 191Wetting angle, 233-234Whiskers, 376-377Wien effects, 98-100Work function, 153-155Working electrode, 252, 292-293

XPS, see Electron spectroscopyX-rays, 336-337

Zero-charge potential, 207-210, 217,231

see also Flatband potentialZwitterion, 71

/S-alumina, 125